Daily Papers

We establish, in general terms, the conditions to be satisfied by a time-fractional approach formulation of the Poisson-Nernst-Planck model in order to guarantee that the total current across the sample be solenoidal, as required by the Maxwell equation. Only in this case the electric impedance of a cell can be determined as the ratio between the applied difference of potential and the current across the cell. We show that in the case of anomalous diffusion, the model predicts for the electric impedance of the cell a constant phase element behaviour in the low frequency region. In the parametric curve of the reactance versus the resistance, the slope coincides with the order of the fractional time derivative.

The classical Coulomb gas model has served as one of the most versatile frameworks in statistical physics, connecting a vast range of phenomena across many different areas. Nonequilibrium generalisations of this model have so far been studied much more scarcely. With the abundance of contemporary research into active and driven systems, one would naturally expect that such generalisations of systems with long-ranged Coulomb-like interactions will form a fertile playground for interesting developments. Here, we present two examples of novel macroscopic behaviour that arise from nonequilibrium fluctuations in long-range interacting systems, namely (1) unscreened long-ranged correlations in strong electrolytes driven by an external electric field and the associated fluctuation-induced forces in the confined Casimir geometry, and (2) out-of-equilibrium critical behaviour in self-chemotactic models that incorporate the particle polarity in the chemotactic response of the cells. Both of these systems have nonlocal Coulomb-like interactions among their constituent particles, namely, the electrostatic interactions in the case of the driven electrolyte, and the chemotactic forces mediated by fast-diffusing signals in the case of self-chemotactic systems. The results presented here hint to the rich phenomenology of nonequilibrium effects that can arise from strong fluctuations in Coulomb interacting systems, and a rich variety of potential future directions, which are discussed.

In this work, we generalize the reaction-diffusion equation in statistical physics, Schr\"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be considered as the fundamental equations in the field of artificial intelligence research. We take finite difference method to discretize NPDE for finding numerical solution, and the basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated. The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented. We believe it is of significance that presented clear physical image of interpretable deep neural networks, which makes it be possible for applying to analog computing device design, and pave the road to physical artificial intelligence.

The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only 2log n+1 items, of its coefficient matrix under a specific set of simple operators, where n is the dimension of the coefficient matrix. This implies that the proposed VQA only needs O(log n) measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.

Physics is based on probabilities as fundamental entities of a mathematical description. Expectation values of observables are computed according to the classical statistical rule. The overall probability distribution for one world covers all times. The quantum formalism arises once one focuses on the evolution of the time-local probabilistic information. Wave functions or the density matrix allow the formulation of a general linear evolution law for classical statistics. The quantum formalism for classical statistics is a powerful tool which allows us to implement for generalized Ising models the momentum observable with the associated Fourier representation. The association of operators to observables permits the computation of expectation values in terms of the density matrix by the usual quantum rule. We show that probabilistic cellular automata are quantum systems in a formulation with discrete time steps and real wave functions. With a complex structure the evolution operator for automata can be expressed in terms of a Hamiltonian involving fermionic creation and annihilation operators. The time-local probabilistic information amounts to a subsystem of the overall probabilistic system which is correlated with its environment consisting of the past and future. Such subsystems typically involve probabilistic observables for which only a probability distribution for their possible measurement values is available. Incomplete statistics does not permit to compute classical correlation functions for arbitrary subsystem-observables. Bell's inequalities are not generally applicable.

In the present work, we study the dynamical evolution of an homogeneous and anisotropic, noncommutative (NC) Bianchi I (BI) model coupled to a radiation perfect fluid. Our first motivation is determining if the present model tends to an homogeneous and isotropic NC Friedmann-Robertson-Walker (FRW) model, during its evolution. In order to simplify our task, we use the Misner parametrization of the BI metric. In terms of that parametrization the BI metric has three metric functions: the scale factor a(t) and the two parameters beta_pm (t), which measure the spatial anisotropy of the model. Our second motivation is trying to describe the present accelerated expansion of the universe using noncommutativity (NCTY). The NCTY is introduced by two nontrivial Poisson brackets between some geometrical as well as matter variables of the model. We recover the description in terms of commutative variables by introducing some variables transformations that depend on the NC parameter. Using those variables transformations, we rewrite the total NC Hamiltonian of the model in terms of commutative variables. From the resulting Hamiltonian, we obtain the dynamical equations for a generic perfect fluid. In order to solve these equations, we restrict our attention to a model where the perfect fluid is radiation. We solve, numerically, these equations and compare the NC solutions to the corresponding commutative ones. The comparison shows that the NC model may be considered as a possible candidate for describing the accelerated expansion of the universe. Finally, we obtain estimates for the NC parameter and compare the main results of the NC BI model coupled to radiation with the same NC BI model coupled to other perfect fluids. As our main result, we show that the solutions, after some time, produce an isotropic universe.

We consider a Schr\"odinger-Poisson system involving a general nonlinearity at critical growth and we prove the existence of positive solutions. The Ambrosetti-Rabinowitz condition is not required. We also study the asymptotics of solutions with respect to a parameter.

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighbouring random sites. The model falls in a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases, some models studied in the literature, like the Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation as advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

The statistical mechanics of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. In this paper, we follow the path set out by Jaynes, including elements added subsequently to that original work, to explore the consequences of the purely statistical point of view. We show how standard methods in the equilibrium theory could have been derived simply from a description of the available problem information. In addition, our presentation leads to novel insights into questions associated with symmetry and non-equilibrium statistical mechanics. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a quantity related to the thermodynamic entropy production is found by considering information loss in non-equilibrium processes. Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complexity by successively adding information to create progressively more complex descriptions of a physical system. Our result is that such statistical mechanical descriptions can be used to create transparent, computable, experimentally-relevant models that may be informed by more detailed atomistic simulations. We also derive a theory for the kinetic behavior of this system, identifying the nonequilibrium `process' free energy functional. The Gibbs relation for this functional is a fluctuation-dissipation theorem applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient driving forces. Based on this work, it is clear that statistical mechanics is a general tool for constructing the relationships between constraints on system information.

This paper studies the properties of the Multiply Iterated Poisson Process (MIPP), a stochastic process constructed by repeatedly time-changing a Poisson process, and its applications in ruin theory. Like standard Poisson processes, MIPPs have exponentially distributed sojourn times (waiting times between jumps). We explicitly derive the probabilities of all possible jump sizes at the first jump and obtain the Laplace transform of the joint distribution of the first jump time and its corresponding jump size. In ruin theory, the classical Cramér-Lundberg model assumes that claims arrive independently according to a Poisson process. In contrast, our model employs an MIPP to allow for clustered arrivals, reflecting real-world scenarios, such as catastrophic events. Under this new framework, we derive the corresponding scale function in closed form, facilitating accurate calculations of the probability of ruin in the presence of clustered claims. These results improve the modeling of extreme risks and have practical implications for insurance solvency assessments, reinsurance pricing, and capital reserve estimation.

The collisionless Boltzmann equation (CBE) is a fundamental equation that governs the dynamics of a broad range of astrophysical systems from space plasma to star clusters and galaxies. It is computationally expensive to integrate the CBE directly in a multi-dimensional phase space, and thus the applications to realistic astrophysical problems have been limited so far. Recently, Todorova & Steijl (2020) proposed an efficient quantum algorithm to solve the CBE with significantly reduced computational complexity. We extend the algorithm to perform quantum simulations of self-gravitating systems, incorporating the method to calculate gravity with the major Fourier modes of the density distribution extracted from the solution-encoding quantum state. Our method improves the dependency of time and space complexities on Nv , the number of grid points in each velocity coordinate, compared to the classical simulation methods. We then conduct some numerical demonstrations of our method. We first run a 1+1 dimensional test calculation of free streaming motion on 64*64 grids using 13 simulated qubits and validate our method. We then perform simulations of Jeans collapse, and compare the result with analytic and linear theory calculations. It will thus allow us to perform large-scale CBE simulations on future quantum computers.

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

A refined semi-holographic non-Fermi liquid model, in which carrier electrons hybridize with operators of a holographic critical sector, has been proposed recently for strange metallic behavior. The model, consistently with effective theory approach, has two couplings whose ratio is related to the doping. We explain the origin of the linear-in-T resistivity and strange metallic behavior as a consequence of the emergence of a universal form of the spectral function which is independent of the model parameters when the ratio of the two couplings take optimal values determined only by the critical exponent. This universal form fits well with photoemission data of copper oxide samples for under/optimal/over-doping with a fixed exponent over a wide range of temperatures. We further obtain a refined Planckian dissipation scenario in which the scattering time τ= f cdot hbar /(k_B T), with f being O(1) at strong coupling, but O(10) at weak coupling.

The observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f*) from Poisson data (y) cannot be effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of f* from y in an inverse problem setting, where (a) the number of unknowns may potentially be larger than the number of observations and (b) f* admits a sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods.

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. These systems generically involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the magnitude of the steady-state probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry and the strength of nonequilibrium transport of measure. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework that estimates the score of this density. We show that the score, together with the microscopic equations of motion, gives direct access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles, spatial regions, and degrees of freedom. To represent the score, we introduce a novel, spatially-local transformer-based network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of interacting active particles undergoing motility-induced phase separation (MIPS). We show that a single instance of our network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Due to wave interference, an ultralight light dark matter halo has stochastic, granular substructures which can scatter stars, leading to the heating of stellar distributions. Studies of this phenomenon have placed lower bounds on the ultralight dark matter mass. In this paper we investigate a number of relevant systematic effects, including: (1) the heating by the central soliton, (2) the self-gravity of the stars, (3) the suppression of heating in a tidally stripped halo, and (4) the tidal field suppression of heating when the stellar cluster is much smaller than the de Broglie wavelength. The first three effects are quantified by studying the dynamics of stellar particles in Schrodinger-Poisson simulations of ultralight dark matter halos, while the last effect is studied using analytic approximations.

We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature beta. In the large N limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form {cal L}_N = sum_{i=1}^N f({bf x}_i), where {bf x}_i's are the positions of the particles and where f({bf x}_i) is a sufficiently regular function. There exists at present standard results for the first and second moments of {cal L}_N in the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of {cal L}_N at large N, when the function f({bf x})=f(|{bf x}|) and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f'(|{bf x}|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d=2 at the special value beta=2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of {cal L}_N.

Understanding equilibration and thermalization in isolated many-body quantum systems is a central challenge in quantum physics. The traditional approach focuses on the study of the full state of the quantum system which, at equilibrium, is best described by the Diagonal Ensemble. Here, we present Observable Statistical Mechanics, a novel paradigm that shifts attention from the full quantum state to the statistics of measurement outcomes. This approach is grounded in the Maximum Observable Entropy Principle, positing that equilibrium measurement statistics tend to maximize observable entropy under conserved average energy. By focusing on accessible measurements, the theory accurately predicts equilibrium probability distributions without needing detailed microscopic information like the energy eigenstates. Extensive numerical experiments on 7 spin-1/2 Hamiltonians demonstrate the broad applicability and robustness of this framework.

In October 2007, a Research Proposal for the University of Sydney, Australia, the author suggested that biovie-physical phenomenon as `electrodynamic dependant biological vision', is governed by relativistic quantum laws and biovision. The phenomenon on the basis of `biovielectroluminescence', satisfies man/microbio/megabio/computer vision (MMMCV), as a robust candidate for physical and visual sciences. The general aim of this addendum is to present a refined text of Sections 1-3 of that proposal and highlighting the contents of its Appendix in form of a `Mechanisms' Section. We then briefly remind in an article aimed for December 2007, by appending two more equations into Section 3, a theoretical II-time scenario as a time model well-proposed for the phenomenon. The time model within the core of the proposal, plays a significant role in emphasizing the principle points on Objectives no. 1-8, Sub-hypothesis 3.1.2, mentioned in Article [arXiv:0710.0410]. It also expresses the time concept in terms of causing quantized energy f(|E|) of time |t|, emit in regard to shortening the probability of particle loci as predictable patterns of particle's un-occurred motion, a solution to Heisenberg's uncertainty principle (HUP) into a simplistic manner. We conclude that, practical frames via a time algorithm to this model, fixates such predictable patterns of motion of scenery bodies onto recordable observation points of a MMMCV system. It even suppresses/predicts superposition phenomena coming from a human subject and/or other bio-subjects for any decision making event, e.g., brainwave quantum patterns based on vision. Maintaining the existential probability of Riemann surfaces of II-time scenarios in the context of biovielectroluminescence, makes motion-prediction a possibility.

We provide a concise framework for generalized ensemble theory through a matrix-based approach. By introducing an observation matrix, any discrete probability distribution, including those for non-equilibrium steady states, can be expressed as a generalized Boltzmann distribution, with observables and conjugate variables as the basis and coordinates in a linear space. In this framework, we identify the minimal sufficient statistics required for inferring the Boltzmann distribution. Furthermore, we show that the Hadamard and Vandermonde matrices are suitable observation matrices for spin systems and random walks. In master equation systems, the probability flux observation matrix facilitates the identification of detailed balance violations. Our findings provide a new approach to developing generalized ensemble theory for non-equilibrium steady-state systems.

Coincident multimessenger observations of cosmic sources can offer numerous benefits, especially when used in the context of synergistic astrophysics. One significant advantage is enhancing the detection significance of separate detectors by correlating their data and assuming joint emission. We have formulated an approach for updating the Bayesian posterior probability of an astrophysical origin, namely p_{rm astro}, relying on multimessenger coincidences assuming an emission model. The description is applicable to any combination of messengers. We demonstrated the formalism for the gravitational waves and high-energy neutrinos case. Applying our method to the public data of candidate coincident high-energy neutrinos with subthreshold gravitational-wave triggers, we found that in the case of highly energetic neutrino coincidences, p_{rm astro} can increase from approximately sim 0.1 to sim 0.9. The amount of improvement depends on the assumed joint emission model. If models are trusted, the marked improvement makes subthreshold detections much more confident. Moreover, the model dependency can also be used to test the consistency of different models. This work is a crucial step toward the goal of uniting all detectors on equal footing into a statistically integrated, Earth-sized observatory for comprehensive multimessenger astrophysics.

This paper describes the 2018 Planck CMB likelihoods, following a hybrid approach similar to the 2015 one, with different approximations at low and high multipoles, and implementing several methodological and analysis refinements. With more realistic simulations, and better correction and modelling of systematics, we can now make full use of the High Frequency Instrument polarization data. The low-multipole 100x143 GHz EE cross-spectrum constrains the reionization optical-depth parameter tau to better than 15% (in combination with with the other low- and high-ell likelihoods). We also update the 2015 baseline low-ell joint TEB likelihood based on the Low Frequency Instrument data, which provides a weaker tau constraint. At high multipoles, a better model of the temperature-to-polarization leakage and corrections for the effective calibrations of the polarization channels (polarization efficiency or PE) allow us to fully use the polarization spectra, improving the constraints on the LambdaCDM parameters by 20 to 30% compared to TT-only constraints. Tests on the modelling of the polarization demonstrate good consistency, with some residual modelling uncertainties, the accuracy of the PE modelling being the main limitation. Using our various tests, simulations, and comparison between different high-ell implementations, we estimate the consistency of the results to be better than the 0.5sigma level. Minor curiosities already present before (differences between ell<800 and ell>800 parameters or the preference for more smoothing of the C_ell peaks) are shown to be driven by the TT power spectrum and are not significantly modified by the inclusion of polarization. Overall, the legacy Planck CMB likelihoods provide a robust tool for constraining the cosmological model and represent a reference for future CMB observations. (Abridged)

In this paper, we discuss that an observable-based single-system Copenhagen and entanglement-based two-system von Neumann measurement protocols in quantum theory can be made equivalent by considering the second part of the two-system scheme to be a Dirac-type negative sea filling up the first system. Based on this equivalence, and by considering the universe as a computational process, the choice of the apparatus state in the two-system protocol can be identified with the choice of the observable in the single-system scheme as negative sea filling up the observable universe. In particular, the measuring party's state is considered to be evolving backwards in time to the big bang as a nondeterministic computational process, which chooses the acceptable path as a time-reversal process of irreversible computation. The suggested model proposes that the prepared microstate of the universe, or reality, corresponds to the observer's choice, therefore, subjective reality. Thus, this effectively provides a specific description of the subjective universe model previously proposed, which is based on the symmetry breakdown between the Schrodinger and the Heisenberg pictures of quantum theory.

I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schr\"odinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological considerations. The theory may have implications for questions of mathematical realism and finitism.

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr\"{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

The friction coefficient of a particle can depend on its position as it does when the particle is near a wall. We formulate the dynamics of particles with such state-dependent friction coefficients in terms of a general Langevin equation with multiplicative noise, whose evaluation requires the introduction of specific rules. Two common conventions, the Ito and the Stratonovich, provide alternative rules for evaluation of the noise, but other conventions are possible. We show the requirement that a particle's distribution function approach the Boltzmann distribution at long times dictates that a drift term must be added to the Langevin equation. This drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise. We explore the consequences of this result in a number examples with spatially varying diffusion coefficients. We also derive path integral representations for arbitrary interpretation of the noise, and use it in a perturbative study of correlations in a simple system.

The dynamical realisation of the equation of state p +rho =0 is studied. A non-pathological dynamics for the perturbations of such a system mimicking a dynamical cosmological constant (DCC) requires to go beyond the perfect fluid paradigm. It is shown that an anisotropic stress must be always present. The Hamiltonian of the system in isolation resembles the one of a Pais-Uhlenbeck oscillator and linear stability requires that it cannot be positive definite. The dynamics of linear cosmological perturbations in a DCC dominated Universe is studied in detail showing that when DCC is minimally coupled to gravity no dramatic instability is present. In contrast to what happens in a cosmological constant dominated Universe, the non-relativistic matter contrast is no longer constant and exhibits an oscillator behaviour at small scales while it grows weakly at large scales. In the gravitational waves sector, at small scales, the amplitude is still suppressed as the inverse power of the scale factor while it grows logarithmically at large scales. Also the vector modes propagate, though no growing mode is found.

We establish a fundamental connection between score-based diffusion models and non-equilibrium thermodynamics by deriving performance limits based on entropy rates. Our main theoretical contribution is a lower bound on the negative log-likelihood of the data that relates model performance to entropy rates of diffusion processes. We numerically validate this bound on a synthetic dataset and investigate its tightness. By building a bridge to entropy rates - system, intrinsic, and exchange entropy - we provide new insights into the thermodynamic operation of these models, drawing parallels to Maxwell's demon and implications for thermodynamic computing hardware. Our framework connects generative modeling performance to fundamental physical principles through stochastic thermodynamics.

Thermophoresis is the migration of a particle due to a thermal gradient. Here, we theoretically uncover the quantum version of thermophoresis. As a proof of principle, we analytically find a thermophoretic force on a trapped quantum particle having three energy levels in Lambda configuration. We then consider a model of N sites, each coupled to its first neighbors and subjected to a local bath at a certain temperature, so as to show numerically how quantum thermophoresis behaves with increasing delocalization of the quantum particle. We discuss how negative thermophoresis and the Dufour effect appear in the quantum regime.

The measurements of the temperature and polarisation anisotropies of the Cosmic Microwave Background (CMB) by the ESA Planck mission have strongly supported the current concordance model of cosmology. However, the latest cosmological data release from ESA Planck mission still has a powerful potential to test new data science algorithms and inference techniques. In this paper, we use advanced Machine Learning (ML) algorithms, such as Neural Networks (NNs), to discern among different underlying cosmological models at the angular power spectra level, using both temperature and polarisation Planck 18 data. We test two different models beyond LambdaCDM: a modified gravity model: the Hu-Sawicki model, and an alternative inflationary model: a feature-template in the primordial power spectrum. Furthermore, we also implemented an interpretability method based on SHAP values to evaluate the learning process and identify the most relevant elements that drive our architecture to certain outcomes. We find that our NN is able to distinguish between different angular power spectra successfully for both alternative models and LambdaCDM. We conclude by explaining how archival scientific data has still a strong potential to test novel data science algorithms that are interesting for the next generation of cosmological experiments.

We study a discrete model for generalized exchange-driven growth in which the particle exchanged between two clusters is not limited to be of size one. This set of models include as special cases the usual exchange-driven growth system and the coagulation-fragmentation system with binary fragmentation. Under reasonable general condition on the rate coefficients we establish the existence of admissible solutions, meaning solutions that are obtained as appropriate limit of solutions to a finite-dimensional truncation of the infinite-dimensional ODE. For these solutions we prove that, in the class of models we call isolated both the total number of particles and the total mass are conserved, whereas in those models we can non-isolated only the mass is conserved. Additionally, under more restrictive growth conditions for the rate equations we obtain uniqueness of solutions to the initial value problems.

This study introduces a novel approach by replacing the traditional perceptron neuron model with a biologically inspired probabilistic meta neuron, where the internal neuron parameters are jointly learned, leading to improved classification accuracy of spiking neural networks (SNNs). To validate this innovation, we implement and compare two SNN architectures: one based on standard leaky integrate-and-fire (LIF) neurons and another utilizing the proposed probabilistic meta neuron model. As a second key contribution, we present a new biologically inspired classification framework that uniquely integrates SNNs with Lempel-Ziv complexity (LZC) a measure closely related to entropy rate. By combining the temporal precision and biological plausibility of SNNs with the capacity of LZC to capture structural regularity, the proposed approach enables efficient and interpretable classification of spatiotemporal neural data, an aspect not addressed in existing works. We consider learning algorithms such as backpropagation, spike-timing-dependent plasticity (STDP), and the Tempotron learning rule. To explore neural dynamics, we use Poisson processes to model neuronal spike trains, a well-established method for simulating the stochastic firing behavior of biological neurons. Our results reveal that depending on the training method, the classifier's efficiency can improve by up to 11.00%, highlighting the advantage of learning additional neuron parameters beyond the traditional focus on weighted inputs alone.

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

All current formulations of thermodynamics invoke some form of coarse-graining or ensembles as the supposed link between their own laws and the microscopic laws of motion. They deal only with ensemble-averages, expectation values, macroscopic limits, infinite heat baths, etc., not with the details of physical variables of individual microscopic systems. They are consistent with the laws of motion for finite systems only in certain approximations, which improve with increasing scale, given various assumptions about initial conditions which are neither specified precisely nor even thought to hold exactly in nature. Here I propose a new formulation of the zeroth, first and second laws, improving upon the axiomatic approach to thermodynamics (Carath\'eodory, 1909; Lieb & Yngvason, 1999), via the principles of the recently proposed constructor theory. Specifically, I provide a non-approximative, scale-independent formulation of 'adiabatic accessibility'; this in turn provides a non-approximative, scale-independent distinction between work and heat and reveals an unexpected connection between information theory and the first law of thermodynamics (not just the second). It also achieves the long-sought unification of the axiomatic approach with Kelvin's.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

It is demonstrated that self-diffusion and shear viscosity data for the TIP4P/Ice water model reported recently [L. Baran, W. Rzysko and L. MacDowell, J. Chem. Phys. {\bf 158}, 064503 (2023)] obey the microscopic version of the Stokes-Einstein relation without the hydrodynamic diameter.

In recent years, diffusion models trained on equilibrium molecular distributions have proven effective for sampling biomolecules. Beyond direct sampling, the score of such a model can also be used to derive the forces that act on molecular systems. However, while classical diffusion sampling usually recovers the training distribution, the corresponding energy-based interpretation of the learned score is often inconsistent with this distribution, even for low-dimensional toy systems. We trace this inconsistency to inaccuracies of the learned score at very small diffusion timesteps, where the model must capture the correct evolution of the data distribution. In this regime, diffusion models fail to satisfy the Fokker--Planck equation, which governs the evolution of the score. We interpret this deviation as one source of the observed inconsistencies and propose an energy-based diffusion model with a Fokker--Planck-derived regularization term to enforce consistency. We demonstrate our approach by sampling and simulating multiple biomolecular systems, including fast-folding proteins, and by introducing a state-of-the-art transferable Boltzmann emulator for dipeptides that supports simulation and achieves improved consistency and efficient sampling. Our code, model weights, and self-contained JAX and PyTorch notebooks are available at https://github.com/noegroup/ScoreMD.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Singular limits for the following indirect signalling chemotaxis system align* \left\{ array{lllllll} \partial_t n = \Delta n - \nabla \cdot (n \nabla c ) & in \Omega\times(0,\infty) , \varepsilon \partial_t c = \Delta c - c + w & in \Omega\times(0,\infty), \varepsilon \partial_t w = \tau \Delta w - w + n & in \Omega\times (0,\infty), \partial_\nu n = \partial_\nu c = \partial_\nu w = 0, &on \partial\Omega\times (0,\infty) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & on \Omega, array \right. align* are investigated. More precisely, we study parabolic-elliptic simplification, or PES, varepsilonto 0^+ with fixed tau>0 up to the critical dimension N=4, and indirect-direct simplification, or IDS, (varepsilon,tau)to (0^+,0^+) up to the critical dimension N=2. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the L^p-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter Ntimes N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies {a_i} and a structurally disordered hopping, we found that each eigenstate is delocalised over N^{2-gamma} sites close in energy |a_j-a_i|leq N^{1-gamma} in agreement with Kravtsov et al, arXiv:1508.01714. Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively ``easy'' PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively ``hard'' PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

Using the supersymmetric method of random matrix theory within the Heidelberg approach framework we provide statistical description of stationary intensity sampled in locations inside an open wave-chaotic cavity, assuming that the time-reversal invariance inside the cavity is fully broken. In particular, we show that when incoming waves are fed via a finite number M of open channels the probability density {cal P}(I) for the single-point intensity I decays as a power law for large intensities: {cal P}(I)sim I^{-(M+2)}, provided there is no internal losses. This behaviour is in marked difference with the Rayleigh law {cal P}(I)sim exp(-I/I) which turns out to be valid only in the limit Mto infty. We also find the joint probability density of intensities I_1, ldots, I_L in L>1 observation points, and then extract the corresponding statistics for the maximal intensity in the observation pattern. For Lto infty the resulting limiting extreme value statistics (EVS) turns out to be different from the classical EVS distributions.

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo & Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called `stochastic interpolants' to bridge any two arbitrary probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent probability density function of the stochastic interpolant is shown to satisfy a first-order transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint immediately leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score of the interpolant density. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also discuss connections with other methods such as score-based diffusion models, stochastic localization processes, probabilistic denoising techniques, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr\"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as the quality and quantity of the training dataset and the network storage, valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate neural networks in general.

A path-integral approach to quantum acoustics is developed here. In contrast to the commonly utilized particle perspective, this emerging field brings forth a long neglected but essential wave paradigm for lattice vibrations. Within the coherent state picture, we formulate a non-Markovian, stochastic master equation that captures the exact dynamics of any system with coupling linear in the bath coordinates and nonlinear in the system coordinates. We further demonstrate the capability of the presented master equation by applying the corresponding procedure to the eminent Fr\"ohlich model. In general, we establish a solid foundation for quantum acoustics as a kindred framework to quantum optics, while paving the way for deeper first-principle explorations of non-perturbative system dynamics driven by lattice vibrations.

X-ray observing facilities, such as the Chandra X-ray Observatory and the eROSITA, have detected millions of astronomical sources associated with high-energy phenomena. The arrival of photons as a function of time follows a Poisson process and can vary by orders-of-magnitude, presenting obstacles for common tasks such as source classification, physical property derivation, and anomaly detection. Previous work has either failed to directly capture the Poisson nature of the data or only focuses on Poisson rate function reconstruction. In this work, we present Poisson Process AutoDecoder (PPAD). PPAD is a neural field decoder that maps fixed-length latent features to continuous Poisson rate functions across energy band and time via unsupervised learning. PPAD reconstructs the rate function and yields a representation at the same time. We demonstrate the efficacy of PPAD via reconstruction, regression, classification and anomaly detection experiments using the Chandra Source Catalog.

In this note, we show that the Local Molecular Field theory of Weeks et. al. can be re-derived as an extremum problem for an approximate Helmholtz free energy. Using the resulting free energy as a classical, fluid density functional yields an implicit solvent method identical in form to the Molecular Density Functional theory of Borgis et. al., but with an explicit formula for the 'ideal' free energy term. This new expression for the ideal free energy term can be computed from all-atom molecular dynamics of a solvent with only short-range interactions. The key hypothesis required to make the theory valid is that all smooth (and hence long-range) energy functions obey Gaussian statistics. This is essentially a random phase approximation for perturbations from a short-range only, 'reference,' fluid. This single hypothesis is enough to prove that the self-consistent LMF procedure minimizes a novel density functional whose 'ideal' free energy is the molecular system under a specific, reference Hamiltonian, as opposed to the non-interacting gas of conventional density functionals. Implementation of this new functional into existing software should be straightforward and robust.

Turbulent magnetic fields are to some extent a universal feature in astrophysical phenomena. Charged particles that encounter these turbulence get on average accelerated according to the so-called second-order Fermi process. However, in most astrophysical environments there are additional competing processes, such as different kinds of first-order energy changes and particle escape, that effect the resulting momentum distribution of the particles. In this work we provide to our knowledge the first semi-analytical solution of the isotropic steady-state momentum diffusion equation including continuous and catastrophic momentum changes that can be applied to any arbitrary astrophysical system of interest. Here, we adopt that the assigned magnetic turbulence is constrained on a finite range and the particle flux vanishes beyond these boundaries. Consequently, we show that the so-called pile-up bump -- that has for some special cases long been established -- is a universal feature of stochastic acceleration that emerges around the momentum chi_{rm eq} where acceleration and continuous loss are in equilibrium if the particle's residence time in the system is sufficient at chi_{rm eq}. In general, the impact of continuous and catastrophic momentum changes plays a crucial role in the shape of the steady-state momentum distribution of the accelerated particles, where simplified unbroken power-law approximations are often not adequate.

We prove existence of weak and strong solutions and uniqueness for a viscous dyadic model driven by additive white noise in time using a path-wise approach. Existence of invariant measures also established and a simple balance relation among the mean rates of energy injection, dissipation and flux is derived and we investigate the asymptotic exponents zeta_{p} of the p-order structure functions.

The Markov entropy decomposition (MED) is a recently-proposed, cluster-based simulation method for finite temperature quantum systems with arbitrary geometry. In this paper, we detail numerical algorithms for performing the required steps of the MED, principally solving a minimization problem with a preconditioned Newton's algorithm, as well as how to extract global susceptibilities and thermal responses. We demonstrate the power of the method with the spin-1/2 XXZ model on the 2D square lattice, including the extraction of critical points and details of each phase. Although the method shares some qualitative similarities with exact-diagonalization, we show the MED is both more accurate and significantly more flexible.

Motivated by the fact that information is encoded and processed by physical systems, the P versus NP problem is examined in terms of physical processes. In particular, we consider P as a class of deterministic, and NP as nondeterministic, polynomial-time physical processes. Based on these identifications, we review a self-reference physical process in quantum theory, which belongs to NP but cannot be contained in P.

Markov jump processes are continuous-time stochastic processes which describe dynamical systems evolving in discrete state spaces. These processes find wide application in the natural sciences and machine learning, but their inference is known to be far from trivial. In this work we introduce a methodology for zero-shot inference of Markov jump processes (MJPs), on bounded state spaces, from noisy and sparse observations, which consists of two components. First, a broad probability distribution over families of MJPs, as well as over possible observation times and noise mechanisms, with which we simulate a synthetic dataset of hidden MJPs and their noisy observation process. Second, a neural network model that processes subsets of the simulated observations, and that is trained to output the initial condition and rate matrix of the target MJP in a supervised way. We empirically demonstrate that one and the same (pretrained) model can infer, in a zero-shot fashion, hidden MJPs evolving in state spaces of different dimensionalities. Specifically, we infer MJPs which describe (i) discrete flashing ratchet systems, which are a type of Brownian motors, and the conformational dynamics in (ii) molecular simulations, (iii) experimental ion channel data and (iv) simple protein folding models. What is more, we show that our model performs on par with state-of-the-art models which are finetuned to the target datasets.

The superfluid phase transition dynamics and associated spontaneous vortex formation with the crossing of the critical temperature in a disk geometry is studied in the framework of the AdS/CFT correspondence by solving the Einstein-Abelian-Higgs model in an AdS_4 black hole. For a slow quench, the vortex density admits a universal scaling law with the cooling rate as predicted by the Kibble-Zurek mechanism (KZM), while for fast quenches, the density shows a universal scaling behavior as a function of the final temperature, that lies beyond the KZM prediction. The vortex number distribution in both the power-law and saturation regimes can be approximated by a normal distribution. However, the study of the universal scaling of the cumulants reveals non-normal features and indicates that vortex statistics in the newborn superfluid is best described by the Poisson binomial distribution, previously predicted in the KZM regime [Phys. Rev. Lett. 124, 240602 (2020)]. This is confirmed by studying the cumulant scalings as a function of the quench time and the quench depth. Our work supports the existence of a universal defect number distribution that accommodates the KZM scaling, its breakdown at fast quenches, and the additional universal scaling laws as a function of the final value of the control parameter.

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

The Eigenstate Thermalization Hypothesis (ETH) has been established as a cornerstone for understanding thermalization in quantum many-body systems. Recently, there has been growing interest in the full ETH, which extends the framework of the conventional ETH and postulates a smooth function to describe the multi-point correlations among matrix elements. Within this framework, free cumulants play a central role, and most previous studies have primarily focused on closed systems. In this paper, we extend the analysis to a system-bath setup, considering both an idealized case with a random-matrix bath and a more realistic scenario where the bath is modeled as a defect Ising chain. In both cases, we uncover a universal scaling of microcanonical free cumulants of system observables with respect to the interaction strength. Furthermore we establish a connection between this scaling behavior and the thermalization dynamics of the thermal free cumulants of corresponding observables.

Gradient descent has proven to be a powerful and effective technique for optimization in numerous machine learning applications. Recent advances in computational neuroscience have shown that learning in standard gradient descent optimization formulation is not consistent with learning in biological systems. This has opened up interesting avenues for building biologically inspired learning techniques. One such approach is inspired by Dale's law, which states that inhibitory and excitatory synapses do not swap roles during the course of learning. The resulting exponential gradient descent optimization scheme leads to log-normally distributed synaptic weights. Interestingly, the density that satisfies the Fokker-Planck equation corresponding to the stochastic differential equation (SDE) with geometric Brownian motion (GBM) is the log-normal density. Leveraging this connection, we start with the SDE governing geometric Brownian motion, and show that discretizing the corresponding reverse-time SDE yields a multiplicative update rule, which surprisingly, coincides with the sampling equivalent of the exponential gradient descent update founded on Dale's law. Furthermore, we propose a new formalism for multiplicative denoising score-matching, subsuming the loss function proposed by Hyvaerinen for non-negative data. Indeed, log-normally distributed data is positive and the proposed score-matching formalism turns out to be a natural fit. This allows for training of score-based models for image data and results in a novel multiplicative update scheme for sample generation starting from a log-normal density. Experimental results on MNIST, Fashion MNIST, and Kuzushiji datasets demonstrate generative capability of the new scheme. To the best of our knowledge, this is the first instance of a biologically inspired generative model employing multiplicative updates, founded on geometric Brownian motion.

We implement physics-informed neural networks (PINNs) to solve the time-independent Schr\"odinger equation for three canonical one-dimensional quantum potentials: an infinite square well, a finite square well, and a finite barrier. The PINN models incorporate trial wavefunctions that exactly satisfy boundary conditions (Dirichlet zeros at domain boundaries), and they optimize a loss functional combining the PDE residual with a normalization constraint. For the infinite well, the ground-state energy is known (E = pi^2 in dimensionless units) and held fixed in training, whereas for the finite well and barrier, the eigenenergy is treated as a trainable parameter. We use fully-connected neural networks with smooth activation functions to represent the wavefunction and demonstrate that PINNs can learn the ground-state eigenfunctions and eigenvalues for these quantum systems. The results show that the PINN-predicted wavefunctions closely match analytical solutions or expected behaviors, and the learned eigenenergies converge to known values. We present training logs and convergence of the energy parameter, as well as figures comparing the PINN solutions to exact results. The discussion addresses the performance of PINNs relative to traditional numerical methods, highlighting challenges such as convergence to the correct eigenvalue, sensitivity to initialization, and the difficulty of modeling discontinuous potentials. We also discuss the importance of the normalization term to resolve the scaling ambiguity of the wavefunction. Finally, we conclude that PINNs are a viable approach for quantum eigenvalue problems, and we outline future directions including extensions to higher-dimensional and time-dependent Schr\"odinger equations.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

This paper concerns a long-range random walk in random environment in dimension 1+1, where the environmental disorder is independent in space but has long-range correlations in time. We prove that two types of rescaled partition functions converge weakly to the Stratonovich solution and the It\^o-Skorohod solution respectively of a fractional stochastic heat equation with multiplicative Gaussian noise which is white in space and colored in time.

We numerically investigate some properties of unbalanced St\"{u}ckelberg holographic superconductors, by considering backreaction effects of fields on the background geometry. More precisely, we study the impacts of the chemical potential mismatch and St\"{u}ckelberg mechanism on the condensation and conductivity types (electrical, spin, mixed, thermo-electric, thermo-spin and thermal conductivity). Our results show that the St\"{u}ckelberg's model parameters C_{alpha} and alpha not only have significant impacts on the phase transition, but also affect the conductivity pseudo-gap and the strength of conductivity fluctuations. Moreover, the effects of these parameters on a system will be gradually reduced as the imbalance grows. We also find that the influence of alpha on the amplitude of conductivity fluctuations depends on the magnitude of the both C_{alpha} and deltamu/mu in the electric and thermal conductivity cases. This results in that increasing alpha can damp the conductivity fluctuations of an unbalanced system in contrast to balanced ones.

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., k-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB

We describe in the context of the particle physics (PP) standard model (SM) `PP-SM' the understanding of the primordial properties and composition of the Universe in the temperature range 130GeV>T>20keV. The Universe evolution is described using FLRW cosmology. We present a global view on particle content across time and describe the different evolution eras using deceleration parameter q. We follow the arrow of time in the expanding and cooling Universe: After the PP-SM heavies (t, h, W, Z) diminish in abundance below Tsimeq 50GeV, the PP-SM plasma in the Universe is governed by the strongly interacting Quark-Gluon content. Once the temperature drops below Tsimeq 150MeV, quarks and gluons hadronize into strongly interacting matter particles. Rapid disappearance of baryonic antimatter completes at T_B=38.2MeV. We study the ensuing disappearance of strangeness and mesons in general. We show that the different eras defined by particle populations are barely separated from each other with abundance of muons fading out just prior to T=O(2.5)MeV, the era of emergence of the free-streaming neutrinos. We discuss the two relevant fundamental constants controlling the decoupling of neutrinos. We subsequently follow the primordial Universe as it passes through the hot dense electron-positron plasma epoch. The high density of positron antimatter disappears near T=20.3keV: Nuclear reactions occur in the presence of a highly mobile and relatively strongly interacting electron-positron plasma phase. We apply plasma theory methods to describe the strong screening effects between heavy dust particle (nucleons). We analyze the paramagnetic characteristics of the electron-positron plasma when exposed to an external primordial magnetic field.

The accessibility of spatially distributed data, enabled by affordable sensors, field, and numerical experiments, has facilitated the development of data-driven solutions for scientific problems, including climate change, weather prediction, and urban planning. Neural Partial Differential Equations (Neural PDEs), which combine deep learning (DL) techniques with domain expertise (e.g., governing equations) for parameterization, have proven to be effective in capturing valuable correlations within spatiotemporal datasets. However, sparse and noisy measurements coupled with modeling approximation introduce aleatoric and epistemic uncertainties. Therefore, quantifying uncertainties propagated from model inputs to outputs remains a challenge and an essential goal for establishing the trustworthiness of Neural PDEs. This work evaluates various Uncertainty Quantification (UQ) approaches for both Forward and Inverse Problems in scientific applications. Specifically, we investigate the effectiveness of Bayesian methods, such as Hamiltonian Monte Carlo (HMC) and Monte-Carlo Dropout (MCD), and a more conventional approach, Deep Ensembles (DE). To illustrate their performance, we take two canonical PDEs: Burger's equation and the Navier-Stokes equation. Our results indicate that Neural PDEs can effectively reconstruct flow systems and predict the associated unknown parameters. However, it is noteworthy that the results derived from Bayesian methods, based on our observations, tend to display a higher degree of certainty in their predictions as compared to those obtained using the DE. This elevated certainty in predictions suggests that Bayesian techniques might underestimate the true underlying uncertainty, thereby appearing more confident in their predictions than the DE approach.

State space models (SSM) have been widely applied for the analysis and visualization of large sequential datasets. Sequential Monte Carlo (SMC) is a very popular particle-based method to sample latent states from intractable posteriors. However, SSM is significantly influenced by the choice of the proposal. Recently Hamiltonian Monte Carlo (HMC) sampling has shown success in many practical problems. In this paper, we propose an SMC augmented by HMC (HSMC) for inference and model learning of nonlinear SSM, which can exempt us from learning proposals and reduce the model complexity significantly. Based on the measure preserving property of HMC, the particles directly generated by transition function can approximate the posterior of latent states arbitrarily well. In order to better adapt to the local geometry of latent space, the HMC is conducted on Riemannian manifold defined by a positive definite metric. In addition, we show that the proposed HSMC method can improve SSMs realized by both Gaussian Processes (GP) and Neural Network (NN).

This paper introduces a novel Hamiltonian-inspired neural network approach to credit scoring, designed to address the challenges of class imbalance and out-of-time (OOT) prediction in financial risk management. Drawing from concepts in Hamiltonian mechanics, we develop a symplectic optimizer and a new loss function to capture the complex dynamics of credit risk evolution. Using the Freddie Mac Single-Family Loan-Level Dataset, we evaluate our model's performance against other machine learning approaches. Our method shows superior discriminative power in OOT scenarios, as measured by the Area Under the Curve (AUC), indicating better ranking ability and robustness to class imbalance. The Hamiltonian-inspired approach shows particular strength in maintaining consistent performance between in-sample and OOT test sets, suggesting improved generalization to future, unseen data. These findings suggest that physics-inspired techniques offer a promising direction for developing more robust and reliable credit scoring models, particularly in uncertain economic situations.

We allow the Higgs field Phi to interact with a dilaton field chi of the background spacetime via the coupling chi^2,Phi^daggerPhi. Upon spontaneous gauge symmetry breaking, the Higgs VEV becomes proportional to chi. While traditionally this linkage is employed to make the Planck mass and particle masses dependent on chi, we present an textit alternative mechanism: the Higgs VEV will be used to construct Planck's constant hbar and speed of light c. Specifically, each open set vicinity of a given point x^* on the spacetime manifold is equipped with a replica of the Glashow-Weinberg-Salam action operating with its own effective values of hbar_* and c_* per hbar_*proptochi^{-1/2}(x^*) and c_*proptochi^{1/2}(x^*), causing these ``fundamental constants'' to vary alongside the dynamical field chi. Moreover, in each open set around x^*, the prevailing value chi(x^*) determines the length and time scales for physical processes occurring in this region as lproptochi^{-1}(x^*) and tauproptochi^{-3/2}(x^*). This leads to an textit anisotropic relation tau^{-1}propto l^{-3/2} between the rate of clocks and the length of rods, resulting in a distinct set of novel physical phenomena. For late-time cosmology, the variation of c along the trajectory of light waves from distant supernovae towards the Earth-based observer necessitates modifications to the Lema\^itre redshift relation and the Hubble law. These modifications are capable of: (1) Accounting for the Pantheon Catalog of SNeIa through a declining speed of light in an expanding Einstein--de Sitter universe, thus avoiding the need for dark energy; (2) Revitalizing Blanchard-Douspis-Rowan-Robinson-Sarkar's CMB power spectrum analysis that bypassed dark energy [A&A 412, 35 (2003)]; and (3) Resolving the H_0 tension without requiring a dynamical dark energy component.

Post-merger gravitational wave echoes provide a unique opportunity to probe the near-horizon structure of astrophysical black holes, that may be modified due to non-perturbative quantum gravity phenomena. However, since the waveform is subject to large theoretical uncertainties, it is necessary to develop model-agnostic search methods for detecting echoes from observational data. A promising strategy is to identify the characteristic quasinormal modes (QNMs) associated with echoes, {\it in frequency space}, which complements existing searches of quasiperiodic pulses in time. In this study, we build upon our previous work targeting these modes by incorporating relative phase information to optimize the Bayesian search algorithm. Using a new phase-marginalized likelihood, the performance can be significantly improved for well-resolved QNMs. This enables an efficient model-agnostic search for QNMs of different shapes by using a simple search template. To demonstrate the robustness of the search algorithm, we construct four complementary benchmarks for the echo waveform that span a diverse range of different theoretical possibilities for the near-horizon structure. We then validate our Bayesian search algorithms by injecting the benchmark models into different realizations of Gaussian noise. Using two types of phase-marginalized likelihoods, we find that the search algorithm can efficiently detect the corresponding QNMs. Therefore, our search strategy provides a concrete Bayesian and model-agnostic approach to "quantum black hole seismology".

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

The Population III.1 theory for supermassive black hole (SMBH) formation predicts a very early (zsim20-25), transient phase, ``The Flash'', of cosmic reionization powered by supermassive stars that are SMBH progenitors. The universe then quickly recombined to become mostly neutral, with this state persisting until galaxies begin to reionize intergalactic gas again at zsim 10. The overall Thomson scattering optical depth, tau, from The Flash has been shown to be tau_{rm PopIII.1}sim0.03, leading to a total tausim0.08-0.09. Such a value, while significantly larger than that previously inferred from {\it Planck} observations of the low-l EE polarization power spectrum of the CMB, can help relieve several ``tensions'' faced by the standard LambdaCDM cosmological model, especially the preference for negative neutrino masses and dynamic dark energy. Here we compute EE power spectra of example models of The Flash. We find that, because of its very high redshift, the contribution to llesssim8 modes is dramatically reduced compared to usual low-z reionization models for the same value of tau, while the power at lgtrsim8 is boosted. Thus the Pop III.1 reionization scenario provides a natural way to increase tau, while remaining closer to the latest CMB low-l polarization observations.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

This study employs a neural network that represents the solution to a Schrödinger bridge problem to perform super-resolution of 2-m temperature in an urban area. Schrödinger bridges generally describe transformations between two data distributions based on diffusion processes. We use a specific Schrödinger-bridge model (SM) that directly transforms low-resolution data into high-resolution data, unlike denoising diffusion probabilistic models (simply, diffusion models; DMs) that generate high-resolution data from Gaussian noise. Low-resolution and high-resolution data were obtained from separate numerical simulations with a physics-based model under common initial and boundary conditions. Compared with a DM, the SM attains comparable accuracy at one-fifth the computational cost, requiring 50 neural-network evaluations per datum for the DM and only 10 for the SM. Furthermore, high-resolution samples generated by the SM exhibit larger variance, implying superior uncertainty quantification relative to the DM. Owing to the reduced computational cost of the SM, our results suggest the feasibility of real-time ensemble micrometeorological prediction using SM-based super-resolution.

This paper develops a comprehensive theoretical framework that imports concepts from stochastic thermodynamics to model price impact and characterize the feasibility of round-trip arbitrage in financial markets. A trading cycle is treated as a non-equilibrium thermodynamic process, where price impact represents dissipative work and market noise plays the role of thermal fluctuations. The paper proves a Financial Second Law: under general convex impact functionals, any round-trip trading strategy yields non-positive expected profit. This structural constraint is complemented by a fluctuation theorem that bounds the probability of profitable cycles in terms of dissipated work and market volatility. The framework introduces a statistical ensemble of trading strategies governed by a Gibbs measure, leading to a free energy decomposition that connects expected cost, strategy entropy, and a market temperature parameter. The framework provides rigorous, testable inequalities linking microstructural impact to macroscopic no-arbitrage conditions, offering a novel physics-inspired perspective on market efficiency. The paper derives explicit analytical results for prototypical trading strategies and discusses empirical validation protocols.

This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world.

We search for dark matter (DM) annihilating subhalos of the Milky Way halo among the Fermi Large Area Telescope (LAT) unassociated sources. We construct, for the first time, a statistical model of the unassociated sources at latitudes above 10 degrees. The latter is built as a combination of both DM annihilation subhalos as well as Galactic and extragalactic astrophysical components. The astrophysical components are constructed based on distributions of associated sources, while the distribution of DM subhalos is derived from Monte Carlo simulations. In this model we take into account the differences in the distributions of associated and unassociated sources including both covariate and prior probability shifts (both being forms of ``dataset shifts''). Previous searches of DM subhalos were based on classify-and-count strategies, while the approach adopted in this work is based on quantification learning, which allows one to determine a well-defined statistical interpretation of the contribution of a population of DM subhalos to the unassociated Fermi-LAT sources. In the bb annihilation channel and for a range of DM masses from 10 GeV to 1 TeV, we don't find a significant contribution from DM subhalos and derive a statistical 95% confidence upper limit on the DM annihilation cross section in this channel. While the derived limits are consistent with previous classify-and-count approaches, our generative statistical model opens new avenues for population studies of Fermi-LAT sources and, more generally, for searches of anomalies on top of backgrounds in presence of statistical and systematic uncertainties.

Although Maxwell discovered the physical laws of electromagnetic waves 160 years ago, how to precisely model the propagation of an RF signal in an electrically large and complex environment remains a long-standing problem. The difficulty is in the complex interactions between the RF signal and the obstacles (e.g., reflection, diffraction, etc.). Inspired by the great success of using a neural network to describe the optical field in computer vision, we propose a neural radio-frequency radiance field, NeRF^2, which represents a continuous volumetric scene function that makes sense of an RF signal's propagation. Particularly, after training with a few signal measurements, NeRF^2 can tell how/what signal is received at any position when it knows the position of a transmitter. As a physical-layer neural network, NeRF^2 can take advantage of the learned statistic model plus the physical model of ray tracing to generate a synthetic dataset that meets the training demands of application-layer artificial neural networks (ANNs). Thus, we can boost the performance of ANNs by the proposed turbo-learning, which mixes the true and synthetic datasets to intensify the training. Our experiment results show that turbo-learning can enhance performance with an approximate 50% increase. We also demonstrate the power of NeRF^2 in the field of indoor localization and 5G MIMO.

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. HOO2024, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a time-reparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.

This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of indifference. It is shown that if a probability measure on an infinite-dimensional function space possesses natural symmetries (invariance under translation, rotation, scaling, and Gaussianity), then the entire solution scheme, including the kernel form, the type of regularization, and the noise parameterization, follows analytically from these postulates. The resulting kernel coincides with a generalized polyharmonic spline; however, unlike existing approaches, it is not chosen empirically but arises as a consequence of the indifference principle. This result provides a theoretical foundation for a broad class of smoothing and interpolation methods, demonstrating their optimality in the absence of a priori information.

In this paper, we develop and implement an efficient asymptotic-preserving (AP) scheme to solve the gas mixture of Boltzmann equations under the disparate mass scaling relevant to the so-called "epochal relaxation" phenomenon. The disparity in molecular masses, ranging across several orders of magnitude, leads to significant challenges in both the evaluation of collision operators and the designing of time-stepping schemes to capture the multi-scale nature of the dynamics. A direct implementation of the spectral method faces prohibitive computational costs as the mass ratio increases due to the need to resolve vastly different thermal velocities. Unlike [I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289], we propose an alternative approach based on proper truncation of asymptotic expansions of the collision operators, which significantly reduces the computational complexity and works well for small varepsilon. By incorporating the separation of three time scales in the model's relaxation process [P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci., 6 (1996), pp. 405-436], we design an AP scheme that captures the specific dynamics of the disparate mass model while maintaining computational efficiency. Numerical experiments demonstrate the effectiveness of the proposed scheme in handling large mass ratios of heavy and light species, as well as capturing the epochal relaxation phenomenon.

We present a probabilistic model for point cloud generation, which is fundamental for various 3D vision tasks such as shape completion, upsampling, synthesis and data augmentation. Inspired by the diffusion process in non-equilibrium thermodynamics, we view points in point clouds as particles in a thermodynamic system in contact with a heat bath, which diffuse from the original distribution to a noise distribution. Point cloud generation thus amounts to learning the reverse diffusion process that transforms the noise distribution to the distribution of a desired shape. Specifically, we propose to model the reverse diffusion process for point clouds as a Markov chain conditioned on certain shape latent. We derive the variational bound in closed form for training and provide implementations of the model. Experimental results demonstrate that our model achieves competitive performance in point cloud generation and auto-encoding. The code is available at https://github.com/luost26/diffusion-point-cloud.

Neural networks are a powerful class of nonlinear functions that can be trained end-to-end on various applications. While the over-parametrization nature in many neural networks renders the ability to fit complex functions and the strong representation power to handle challenging tasks, it also leads to highly correlated neurons that can hurt the generalization ability and incur unnecessary computation cost. As a result, how to regularize the network to avoid undesired representation redundancy becomes an important issue. To this end, we draw inspiration from a well-known problem in physics -- Thomson problem, where one seeks to find a state that distributes N electrons on a unit sphere as evenly as possible with minimum potential energy. In light of this intuition, we reduce the redundancy regularization problem to generic energy minimization, and propose a minimum hyperspherical energy (MHE) objective as generic regularization for neural networks. We also propose a few novel variants of MHE, and provide some insights from a theoretical point of view. Finally, we apply neural networks with MHE regularization to several challenging tasks. Extensive experiments demonstrate the effectiveness of our intuition, by showing the superior performance with MHE regularization.

This article deals with the following fractional (p,q)-Choquard equation with exponential growth of the form: $varepsilon^{ps}(-Delta)_{p}^{s}u+varepsilon^{qs}(-Delta)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=varepsilon^{mu-N}[|x|^{-mu}*F(u)]f(u) in R^N, where s\in (0,1), \varepsilon>0 is a parameter, 2\leq p=N{s}<q, and 0<\mu<N. The nonlinear function f has an exponential growth at infinity and the continuous potential function Z satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for \varepsilon>0$ small enough. In a certain sense, we generalize some previously known results.

We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions which are accessible by samples. Our algorithm is based on the saddle point reformulation of the dynamic version of EOT which is known as the Schr\"odinger Bridge problem. In contrast to the prior methods for large-scale EOT, our algorithm is end-to-end and consists of a single learning step, has fast inference procedure, and allows handling small values of the entropy regularization coefficient which is of particular importance in some applied problems. Empirically, we show the performance of the method on several large-scale EOT tasks. https://github.com/ngushchin/EntropicNeuralOptimalTransport

We present a neural flow wavefunction, Gauge-Fermion FlowNet, and use it to simulate 2+1D lattice compact quantum electrodynamics with finite density dynamical fermions. The gauge field is represented by a neural network which parameterizes a discretized flow-based transformation of the amplitude while the fermionic sign structure is represented by a neural net backflow. This approach directly represents the U(1) degree of freedom without any truncation, obeys Guass's law by construction, samples autoregressively avoiding any equilibration time, and variationally simulates Gauge-Fermion systems with sign problems accurately. In this model, we investigate confinement and string breaking phenomena in different fermion density and hopping regimes. We study the phase transition from the charge crystal phase to the vacuum phase at zero density, and observe the phase seperation and the net charge penetration blocking effect under magnetic interaction at finite density. In addition, we investigate a magnetic phase transition due to the competition effect between the kinetic energy of fermions and the magnetic energy of the gauge field. With our method, we further note potential differences on the order of the phase transitions between a continuous U(1) system and one with finite truncation. Our state-of-the-art neural network approach opens up new possibilities to study different gauge theories coupled to dynamical matter in higher dimensions.

Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general. The particle-based, rapid incremental smoother PaRIS is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models. Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC). PARIS has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees. Still, being based on self-normalised importance sampling, the PaRIS estimator is biased. Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs PPG sampler, which can be viewed as a PaRIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. Our second contribution is to apply PPG in a learning framework, covering MLE and MSC as special examples. In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao--Blackwellization of PPG. These are the first non-asymptotic results of this kind in this setting. We illustrate our theoretical results with numerical experiments supporting our claims.

We present a neural network emulator to constrain the thermal parameters of the intergalactic medium (IGM) at 5.4z6.0 using the Lyman-displaystylealpha (Lydisplaystylealpha) forest flux auto-correlation function. Our auto-differentiable JAX-based framework accelerates the surrogate model generation process using approximately 100 sparsely sampled Nyx hydrodynamical simulations with varying combinations of thermal parameters, i.e., the temperature at mean density T_{{0}}, the slope of the temperaturedisplaystyle-density relation displaystylegamma, and the mean transmission flux langle{F}{rangle}. We show that this emulator has a typical accuracy of 1.0% across the specified redshift range. Bayesian inference of the IGM thermal parameters, incorporating emulator uncertainty propagation, is further expedited using NumPyro Hamiltonian Monte Carlo. We compare both the inference results and computational cost of our framework with the traditional nearest-neighbor interpolation approach applied to the same set of mock Lyalpha flux. By examining the credibility contours of the marginalized posteriors for T_{{0}},gamma,and{langle}{F}{rangle} obtained using the emulator, the statistical reliability of measurements is established through inference on 100 realistic mock data sets of the auto-correlation function.

We study the dynamics of gravitons in a squeezed vacuum state in a thermal radiation background. Unlike traditional treatments that rely on the Boltzmann equation, we employ the Heisenberg equation and average it over general quantum states. In contrast to the usual Boltzmann-based descriptions, our approach captures the subtleties arising from quantum coherence in different number eigenstates, which is essential for soft graviton modes in the squeezed vacuum state. Our new method successfully reproduces the previous one-loop results within the in-in formalism when the expansion parameter is small and deviates significantly as the parameter increases, indicating that our results extend beyond the one-loop in-in formalism. We examine the implications of graviton emission effects stimulated by quantum coherence in both flat and expanding backgrounds. In the flat background, it is found that backreaction of radiation on the spacetime dynamics is crucial for significant stimulated emission. In the expanding background, to avoid the subtleties associated with superhorizon modes, we investigate the effect of emission within the horizon immediately after reheating and find a significant effect. We examined the IR graviton evolution from a symmetry perspective and propose a regularization prescription to eliminate the secular growth problem.

Poisson noise suppression is an important preprocessing step in several applications, such as medical imaging, microscopy, and astronomical imaging. In this work, we propose a novel patch-wise Poisson noise removal strategy, in which the MMSE estimator is utilized in order to produce the denoising result for each image patch. Fast and accurate computation of the MMSE estimator is carried out using k-d tree search followed by search in the K-nearest neighbor graph. Our experiments show that the proposed method is the preferable choice for low signal-to-noise ratios.

Energy-based models (EBMs) are known in the Machine Learning community for decades. Since the seminal works devoted to EBMs dating back to the noughties, there have been a lot of efficient methods which solve the generative modelling problem by means of energy potentials (unnormalized likelihood functions). In contrast, the realm of Optimal Transport (OT) and, in particular, neural OT solvers is much less explored and limited by few recent works (excluding WGAN-based approaches which utilize OT as a loss function and do not model OT maps themselves). In our work, we bridge the gap between EBMs and Entropy-regularized OT. We present a novel methodology which allows utilizing the recent developments and technical improvements of the former in order to enrich the latter. From the theoretical perspective, we prove generalization bounds for our technique. In practice, we validate its applicability in toy 2D and image domains. To showcase the scalability, we empower our method with a pre-trained StyleGAN and apply it to high-res AFHQ 512times 512 unpaired I2I translation. For simplicity, we choose simple short- and long-run EBMs as a backbone of our Energy-guided Entropic OT approach, leaving the application of more sophisticated EBMs for future research. Our code is available at: https://github.com/PetrMokrov/Energy-guided-Entropic-OT

Diffusion models may be formulated as a time-indexed sequence of energy-based models, where the score corresponds to the negative gradient of an energy function. As opposed to learning the score directly, an energy parameterization is attractive as the energy itself can be used to control generation via Monte Carlo samplers. Architectural constraints and training instability in energy parameterized models have so far yielded inferior performance compared to directly approximating the score or denoiser. We address these deficiencies by introducing a novel training regime for the energy function through distillation of pre-trained diffusion models, resembling a Helmholtz decomposition of the score vector field. We further showcase the synergies between energy and score by casting the diffusion sampling procedure as a Feynman Kac model where sampling is controlled using potentials from the learnt energy functions. The Feynman Kac model formalism enables composition and low temperature sampling through sequential Monte Carlo.

Practitioners frequently aim to infer an unobserved population trajectory using sample snapshots at multiple time points. For instance, in single-cell sequencing, scientists would like to learn how gene expression evolves over time. But sequencing any cell destroys that cell. So we cannot access any cell's full trajectory, but we can access snapshot samples from many cells. Stochastic differential equations are commonly used to analyze systems with full individual-trajectory access; since here we have only sample snapshots, these methods are inapplicable. The deep learning community has recently explored using Schr\"odinger bridges (SBs) and their extensions to estimate these dynamics. However, these methods either (1) interpolate between just two time points or (2) require a single fixed reference dynamic within the SB, which is often just set to be Brownian motion. But learning piecewise from adjacent time points can fail to capture long-term dependencies. And practitioners are typically able to specify a model class for the reference dynamic but not the exact values of the parameters within it. So we propose a new method that (1) learns the unobserved trajectories from sample snapshots across multiple time points and (2) requires specification only of a class of reference dynamics, not a single fixed one. In particular, we suggest an iterative projection method inspired by Schr\"odinger bridges; we alternate between learning a piecewise SB on the unobserved trajectories and using the learned SB to refine our best guess for the dynamics within the reference class. We demonstrate the advantages of our method via a well-known simulated parametric model from ecology, simulated and real data from systems biology, and real motion-capture data.

The AKSZ formalism is a construction of topological field theories where the target spaces are differential graded symplectic manifolds. In this paper, we describe an analogue of the AKSZ formalism where the target spaces are differential graded contact manifolds. We show that the space of fields inherits a weak contact structure, and we construct a solution to the analogue of the classical master equation, defined via the Jacobi bracket. In the n=1 case, we recover the Jacobi sigma model, and in the n=2 case, we obtain three-dimensional topological field theories associated to Courant-Jacobi algebroids.

We use new cosmic microwave background (CMB) primary temperature and polarization anisotropy measurements from the Atacama Cosmology Telescope (ACT) Data Release 6 (DR6) to test foundational assumptions of the standard cosmological model and set constraints on extensions to it. We derive constraints from the ACT DR6 power spectra alone, as well as in combination with legacy data from Planck. To break geometric degeneracies, we include ACT and Planck CMB lensing data and baryon acoustic oscillation data from DESI Year-1, and further add supernovae measurements from Pantheon+ for models that affect the late-time expansion history. We verify the near-scale-invariance (running of the spectral index d n_s/dln k = 0.0062 pm 0.0052) and adiabaticity of the primordial perturbations. Neutrino properties are consistent with Standard Model predictions: we find no evidence for new light, relativistic species that are free-streaming (N_{rm eff} = 2.86 pm 0.13, which combined with external BBN data becomes N_{rm eff} = 2.89 pm 0.11), for non-zero neutrino masses (sum m_nu < 0.082 eV at 95% CL), or for neutrino self-interactions. We also find no evidence for self-interacting dark radiation (N_{rm idr} < 0.134), early-universe variation of fundamental constants, early dark energy, primordial magnetic fields, or modified recombination. Our data are consistent with standard BBN, the FIRAS-inferred CMB temperature, a dark matter component that is collisionless and with only a small fraction allowed as axion-like particles, a cosmological constant, and the late-time growth rate predicted by general relativity. We find no statistically significant preference for a departure from the baseline LambdaCDM model. In general, models introduced to increase the Hubble constant or to decrease the amplitude of density fluctuations inferred from the primary CMB are not favored by our data.

I review two classes of strong coupling problems in condensed matter physics, and describe insights gained by application of the AdS/CFT correspondence. The first class concerns non-zero temperature dynamics and transport in the vicinity of quantum critical points described by relativistic field theories. I describe how relativistic structures arise in models of physical interest, present results for their quantum critical crossover functions and magneto-thermoelectric hydrodynamics. The second class concerns symmetry breaking transitions of two-dimensional systems in the presence of gapless electronic excitations at isolated points or along lines (i.e. Fermi surfaces) in the Brillouin zone. I describe the scaling structure of a recent theory of the Ising-nematic transition in metals, and discuss its possible connection to theories of Fermi surfaces obtained from simple AdS duals.

Energy-Based Models (EBMs) have emerged as a powerful framework in the realm of generative modeling, offering a unique perspective that aligns closely with principles of statistical mechanics. This review aims to provide physicists with a comprehensive understanding of EBMs, delineating their connection to other generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Normalizing Flows. We explore the sampling techniques crucial for EBMs, including Markov Chain Monte Carlo (MCMC) methods, and draw parallels between EBM concepts and statistical mechanics, highlighting the significance of energy functions and partition functions. Furthermore, we delve into state-of-the-art training methodologies for EBMs, covering recent advancements and their implications for enhanced model performance and efficiency. This review is designed to clarify the often complex interconnections between these models, which can be challenging due to the diverse communities working on the topic.

Neural network (NN) potentials promise highly accurate molecular dynamics (MD) simulations within the computational complexity of classical MD force fields. However, when applied outside their training domain, NN potential predictions can be inaccurate, increasing the need for Uncertainty Quantification (UQ). Bayesian modeling provides the mathematical framework for UQ, but classical Bayesian methods based on Markov chain Monte Carlo (MCMC) are computationally intractable for NN potentials. By training graph NN potentials for coarse-grained systems of liquid water and alanine dipeptide, we demonstrate here that scalable Bayesian UQ via stochastic gradient MCMC (SG-MCMC) yields reliable uncertainty estimates for MD observables. We show that cold posteriors can reduce the required training data size and that for reliable UQ, multiple Markov chains are needed. Additionally, we find that SG-MCMC and the Deep Ensemble method achieve comparable results, despite shorter training and less hyperparameter tuning of the latter. We show that both methods can capture aleatoric and epistemic uncertainty reliably, but not systematic uncertainty, which needs to be minimized by adequate modeling to obtain accurate credible intervals for MD observables. Our results represent a step towards accurate UQ that is of vital importance for trustworthy NN potential-based MD simulations required for decision-making in practice.

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

The approximation of Partial Differential Equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive biases of neural networks. However, they usually require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, the fixed positions of the mesh parameters restrict their flexibility, making it challenging to accurately approximate complex PDEs. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

We describe nonparametric deconvolution models (NDMs), a family of Bayesian nonparametric models for collections of data in which each observation is the average over the features from heterogeneous particles. For example, these types of data are found in elections, where we observe precinct-level vote tallies (observations) of individual citizens' votes (particles) across each of the candidates or ballot measures (features), where each voter is part of a specific voter cohort or demographic (factor). Like the hierarchical Dirichlet process, NDMs rely on two tiers of Dirichlet processes to explain the data with an unknown number of latent factors; each observation is modeled as a weighted average of these latent factors. Unlike existing models, NDMs recover how factor distributions vary locally for each observation. This uniquely allows NDMs both to deconvolve each observation into its constituent factors, and also to describe how the factor distributions specific to each observation vary across observations and deviate from the corresponding global factors. We present variational inference techniques for this family of models and study its performance on simulated data and voting data from California. We show that including local factors improves estimates of global factors and provides a novel scaffold for exploring data.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

In this paper I apply newly-proposed information-theoretic principles to thermodynamic work extraction. I show that if it is possible to extract work deterministically from a physical system prepared in any one of a set of states, then those states must be distinguishable from one another. This result is formulated independently of scale and of particular dynamical laws; it also provides a novel connection between thermodynamics and information theory, established via the law of conservation of energy (rather than the second law of thermodynamics). Albeit compatible with these conclusions, existing thermodynamics approaches cannot provide a result of such generality, because they are scale-dependent (relying on ensembles or coarse-graining) or tied to particular dynamical laws. This paper thus provides a broader foundation for thermodynamics, with implications for the theory of von Neumann's universal constructor

We model a compact black hole-accretion disk system in the collapsar scenario with full transport, frequency dependent, general relativistic radiation magnetohydrodynamics. We examine whether or not winds from a collapsar disk can undergo rapid neutron capture (r-process) nucleosynthesis and significantly contribute to solar r-process abundances. We find the inclusion of accurate transport has significant effects on outflows, raising the electron fraction above Y_{rm e} sim 0.3 and preventing third peak r-process material from being synthesized. We analyze the time-evolution of neutrino processes and electron fraction in the disk and present a simple one-dimensional model for the vertical structure that emerges. We compare our simulation to semi-analytic expectations and argue that accurate neutrino transport and realistic initial and boundary conditions are required to capture the dynamics and nucleosynthetic outcome of a collapsar.

Long-term forecasting of chaotic systems from short-term observations remains a fundamental and underexplored challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Existing approaches often rely on long-term training data or focus on short-term sequence correlations, struggling to maintain predictive stability and dynamical coherence over extended horizons. We propose PhyxMamba, a novel framework that integrates a Mamba-based state-space model with physics-informed principles to capture the underlying dynamics of chaotic systems. By reconstructing the attractor manifold from brief observations using time-delay embeddings, PhyxMamba extracts global dynamical features essential for accurate forecasting. Our generative training scheme enables Mamba to replicate the physical process, augmented by multi-token prediction and attractor geometry regularization for physical constraints, enhancing prediction accuracy and preserving key statistical invariants. Extensive evaluations on diverse simulated and real-world chaotic systems demonstrate that PhyxMamba delivers superior long-term forecasting and faithfully captures essential dynamical invariants from short-term data. This framework opens new avenues for reliably predicting chaotic systems under observation-scarce conditions, with broad implications across climate science, neuroscience, epidemiology, and beyond. Our code is open-source at https://github.com/tsinghua-fib-lab/PhyxMamba.

Galaxy clusters selected based on overdensities of galaxies in photometric surveys provide the largest cluster samples. Yet modeling the selection function of such samples is complicated by non-cluster members projected along the line of sight (projection effects) and the potential detection of unvirialized objects (contamination). We empirically constrain the magnitude of these effects by cross-matching galaxy clusters selected in the Dark Energy survey data with the \rdmpr, algorithm with significant detections in three South Pole Telescope surveys (SZ, pol-ECS, pol-500d). For matched clusters, we augment the \rdmpr,catalog by the SPT detection significance. For unmatched objects we use the SPT detection threshold as an upper limit on the SZe signature. Using a Bayesian population model applied to the collected multi-wavelength data, we explore various physically motivated models to describe the relationship between observed richness and halo mass. Our analysis reveals the limitations of a simple lognormal scatter model in describing the data. We rule out significant contamination by unvirialized objects at the high-richness end of the sample. While dedicated simulations offer a well-fitting calibration of projection effects, our findings suggest the presence of redshift-dependent trends that these simulations may not have captured. Our findings highlight that modeling the selection function of optically detected clusters remains a complicated challenge, requiring a combination of simulation and data-driven approaches.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

This study examines the performance of finite spin quantum batteries (QBs) using Heisenberg spin models with Dzyaloshinsky-Moriya (DM) and Kaplan--Shekhtman--Entin-Wohlman--Aharony (KSEA) interactions. The QBs are modeled as interacting quantum spins in local inhomogeneous magnetic fields, inducing variable Zeeman splitting. We derive analytical expressions for the maximal extractable work, ergotropy and the capacity of QBs, as recently examined by Yang et al. [Phys. Rev. Lett. 131, 030402 (2023)]. These quantities are analytically linked through certain quantum correlations, as posited in the aforementioned study. Different Heisenberg spin chain models exhibit distinct behaviors under varying conditions, emphasizing the importance of model selection for optimizing QB performance. In antiferromagnetic (AFM) systems, maximum ergotropy occurs with a Zeeman splitting field applied to either spin, while ferromagnetic (FM) systems benefit from a uniform Zeeman field. Temperature significantly impacts QB performance, with ergotropy in the AFM case being generally more robust against temperature increases compared to the FM case. Incorporating DM and KSEA couplings can significantly enhance the capacity and ergotropy extraction of QBs. However, there exists a threshold beyond which additional increases in these interactions cause a sharp decline in capacity and ergotropy. This behavior is influenced by temperature and quantum coherence, which signal the occurrence of a sudden phase transition. The resource theory of quantum coherence proposed by Baumgratz et al. [Phys. Rev. Lett. 113, 140401 (2014)] plays a crucial role in enhancing ergotropy and capacity. However, ergotropy is limited by both the system's capacity and the amount of coherence. These findings support the theoretical framework of spin-based QBs and may benefit future research on quantum energy storage devices.

Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational quantum learning models to predict potential energy surfaces and atomic forces from ab initio training data. However, the trainability and scalability of such models are still limited, due to both theoretical and practical barriers. Inspired by recent developments in geometric classical and quantum machine learning, here we design quantum neural networks that explicitly incorporate, as a data-inspired prior, an extensive set of physically relevant symmetries. We find that our invariant quantum learning models outperform their more generic counterparts on individual molecules of growing complexity. Furthermore, we study a water dimer as a minimal example of a system with multiple components, showcasing the versatility of our proposed approach and opening the way towards larger simulations. Our results suggest that molecular force fields generation can significantly profit from leveraging the framework of geometric quantum machine learning, and that chemical systems represent, in fact, an interesting and rich playground for the development and application of advanced quantum machine learning tools.

The dynamics of Einstein-conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman-Robertson--Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (`Small Bang'), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (`Big Bang'). Thus, in the present EccH model there is a new kind of physical spacetime singularity (`Small Bang'). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout-Englert-Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

In this paper we consider a particular class of solutions of the Rayleigh-Boltzmann equation, known in the nonlinear setting as homoenergetic solutions, which have the form gleft( x,v,t right) =fleft( v-Lleft( tright)x,tright) where the matrix L(t) describes a shear flow deformation. We began this analysis in [22] where we rigorously proved the existence of a stationary non-equilibrium solution and established the different behaviour of the solutions for small and large values of the shear parameter, for cut-off collision kernels with homogeneity parameter 0leq gamma <1, including Maxwell molecules and hard potentials. In this paper, we concentrate in the case where the deformation term dominates the collision term for large times (hyperbolic-dominated regime). This occurs for collision kernels with gamma < 0 and in particular we focus on gamma in (-1,0). In such a hyperbolic-dominated regime, it appears challenging to provide a clear description of the long-term asymptotics of the solutions. Here we present a formal analysis of the long-time asymptotics for the distribution of velocities and provide the explicit form for the asymptotic profile. Additionally, we discuss the different asymptotic behaviour expected in the case of homogeneity gamma < -1. Furthermore, we provide a probabilistic interpretation describing a stochastic process consisting in a combination of collisions and shear flows. The tagged particle velocity {v(t)}_{tgeq 0} is a Markov process that arises from the combination of free flights in a shear flow along with random jumps caused by collisions.