# A Deep Dive into Machine Learning Density Functional Theory for Materials
Science and Chemistry

L. Fiedler<EMAIL_ADDRESS>Center for Advanced Systems Understanding
(CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-Rossendorf,
D-01328 Dresden, Germany  K. Shah<EMAIL_ADDRESS>Center for Advanced Systems
Understanding (CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-
Rossendorf, D-01328 Dresden, Germany  M. Bussmann<EMAIL_ADDRESS>Center
for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany  A. Cangi
<EMAIL_ADDRESS>Center for Advanced Systems Understanding (CASUS), D-02826
Görlitz, Germany  Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden,
Germany

(August 27, 2024)

###### Abstract

With the growth of computational resources, the scope of electronic structure
simulations has increased greatly. Artificial intelligence and robust data
analysis hold the promise to accelerate large-scale simulations and their
analysis to hitherto unattainable scales. Machine learning is a rapidly
growing field for the processing of such complex datasets. It has recently
gained traction in the domain of electronic structure simulations, where
density functional theory takes the prominent role of the most widely used
electronic structure method. Thus, DFT calculations represent one of the
largest loads on academic high-performance computing systems across the world.
Accelerating these with machine learning can reduce the resources required and
enables simulations of larger systems. Hence, the combination of density
functional theory and machine learning has the potential to rapidly advance
electronic structure applications such as in-silico materials discovery and
the search for new chemical reaction pathways. We provide the theoretical
background of both density functional theory and machine learning on a
generally accessible level. This serves as the basis of our comprehensive
review including research articles up to December 2020 in chemistry and
materials science that employ machine-learning techniques. In our analysis, we
categorize the body of research into main threads and extract impactful
results. We conclude our review with an outlook on exciting research
directions in terms of a citation analysis.

††preprint: ml-dft-review/1.0

###### Contents

  1. I Introduction
  2. II Theoretical Background
    1. II.1 Electronic Structure Theory
      1. II.1.1 Exact framework
      2. II.1.2 Practical methods
    2. II.2 Machine learning
      1. II.2.1 Introduction
      2. II.2.2 Supervised Learning Models
  3. III Machine-learning in Materials Science and Chemistry
    1. III.1 Property mappings
      1. III.1.1 Chemical Reactions
      2. III.1.2 Discovery of Novel, Stable Materials
      3. III.1.3 Electric Properties and Photovoltaics
      4. III.1.4 Elastic and Structural Properties
      5. III.1.5 Quantum Chemical Information
      6. III.1.6 Magnetic, Thermal, and Energetic Properties
    2. III.2 Interatomic Potentials
      1. III.2.1 Neural Network Potentials
      2. III.2.2 Gaussian Approximation Potentials
      3. III.2.3 Moment Tensor Potentials
      4. III.2.4 Spectral Neighborhood Analysis and Other Potentials
      5. III.2.5 Technical Aspects for the Constructing of Potentials
    3. III.3 Learning Electronic Structures
    4. III.4 Other Approaches
    5. III.5 Technical Aspects
  4. IV Discussion
    1. IV.1 Citation Analysis
    2. IV.2 Conclusions and Outlook

##  I Introduction

Electronic structure theory calculations enable the understanding of matter on
the quantum level and complement experimental studies both in material science
and chemistry. They play a central role in solving pressing scientific and
technological problems. Large-scale electronic structure simulations have in
turn been enabled by the advent of modern, high-performance computational
resources. Yet, the ever increasing demand for accurate first-principles data
renders even the most efficient simulation codes infeasible. On the other
hand, the use of data-driven machine-learning methods has grown rapidly across
a number of research fields. Such methods are increasingly gaining importance,
as they are utilized to accelerate, replace, or improve traditional electronic
structure theory methods.

The computational backbone of electronic structure workflows is often density
functional theory (DFT). While DFT provides a convenient balance between
computational cost and accuracy, enormous speed-ups can be achieved when
combined with machine learning (ML). In the following, we provide an overview
over recent research efforts that accelerate materials science with the aid of
ML and tackle different aspects of the combined ML-DFT workflow outlined
schematically in Fig. 1. To that end, we compiled a database of over 300
research articles including the most recent publications up to December 2020.
In Sec. II we introduce the necessary formalism and methodologies. Sec. III is
the centerpiece of our manuscript, where we provide an in-depth review of the
compiled publications according to the most relevant research lines and
highlight the most prominent applications. Finally, in Sec. IV, we carry out a
citation analysis and point to trends of future research.

Figure 1: Schematic overview of the general workflow combining ML and DFT for
material science or chemistry applications.

##  II Theoretical Background

###  II.1 Electronic Structure Theory

Developing practical methods for calculating the electronic structure of
materials is a topic of active research. In the following, we provide a short
overview of the basic formalism with a focus on DFT. We work within atomic
units throughout, where $\hbar=m_{e}=e^{2}=1$, such that energies are
expressed in Hartrees and length in Bohr radii.

####  II.1.1 Exact framework

The most common theoretical framework for describing thermodynamic materials
properties from first principles is within the scope of non-relativistic
quantum mechanics. It provides a systematic representation of $N_{e}$
electrons with collective coordinates
$\underline{\bm{r}}=\\{\bm{r}_{1},\dots,\bm{r}_{N_{e}}\\}$ that are coupled to
$N_{i}$ ions with collective coordinates
$\underline{\bm{R}}=\\{\bm{R}_{1},\dots,\bm{R}_{N_{i}}\\}$, where
$\bm{r}_{j}\in\mathbb{R}^{3}$ refers to the position of the $j$th electron,
while $\bm{R}_{\alpha}\in\mathbb{R}^{3}$ denotes the position of the
$\alpha$th ion. Note that for sake of brevity we do not take spins into
account. The physics in this framework is governed by the Hamiltonian [1]

$\displaystyle\hat{H}^{BO}(\underline{\bm{r}};\underline{\bm{R}})$
$\displaystyle=\hat{T}^{e}(\underline{\bm{r}})+\hat{V}^{ee}(\underline{\bm{r}})+\hat{V}^{ei}(\underline{\bm{r}};\underline{\bm{R}})+E^{ii}(\underline{\bm{R}})\;,$
(1)

where $\hat{T}^{e}$ refers to the kinetic energy of the electrons, while the
Coulomb operators, $\hat{V}^{ei}$ and $\hat{V}^{ee}$ account for the electron-
ion and electron-electron interactions. In this equation, the Born-Oppenheimer
approximation [2] is employed to reduce the computational complexity of the
underlying problem, i.e., the coupled electron-ion problem is separated and
the ion-ion interaction amounts to a constant shift in energy $E^{ii}$. This
approximation is valid as long as the motion of the ions happens on a much
larger time scale than the motion of the electrons. Employing this Hamiltonian
in the Schrödinger equation

$\hat{H}^{BO}(\underline{\bm{r}};\underline{\bm{R}})\Psi(\underline{\bm{r}};\underline{\bm{R}})=E\Psi(\underline{\bm{r}};\underline{\bm{R}})\;,$
(2)

with the many-electron wavefunction $\Psi$, which depends on the electronic
coordinates and only parametrically on the ionic positions, provides the basis
for the electronic structure of matter. It therefore enables quantitative
predictions of physical phenomena in materials science. Its solution yields
the electronic ground state based on which a vast amount of materials
properties can be computed. Most importantly, the electronic ground-state
energy can be identified as

$E=\bra{\Psi}\hat{H}^{BO}\ket{\Psi}\;.$ (3)

Other important quantities include molecular or crystal structures [3], charge
densities [4], cohesive energies [5], elastic properties [6], vibrational
properties [7], magnetic properties [8, 9], dielectric susceptibilities [10],
magnetic susceptibilities [11], phase transitions [12], HOMO/LUMO energies
[13, 14, 15], bond dissociations [16], enthalpies of formation [16],
ionization potentials [17], electron affinities [17], band gaps [18, 19], and
the equation of state [20].

####  II.1.2 Practical methods

The multitude of practical electronic structure methods employ different
approximations to overcome the underlying complexity of the problem. In the
following concise overview of these methods, we always work within the Born-
Oppenheimer approximation and suppress denoting the parametric dependence on
$\underline{\bm{R}}$.

###### Density Functional Theory

DFT is a very popular method for carrying out electronic structure
calculations, as it balances acceptable accuracy with reasonable computational
cost. The central quantity is the electronic density $n(\bm{r})$. The
Hohenberg-Kohn theorems [21] guarantee a one-to-one correspondence between the
electronic density and the external potential, e.g., the electron-ion
potential $\hat{V}^{ei}(\underline{\bm{r}})$. This means every property of
interest can be determined as a functional of the density.

Practical DFT calculations usually rely on the Kohn-Sham ansatz [22]. Here,
the system of interacting electrons is replaced by an auxiliary system of non-
interacting electrons constrained to reproduce the electronic density of the
interacting system. This is achieved in terms of the Kohn-Sham equations

$\displaystyle\left[-\frac{1}{2}\nabla^{2}+v_{\mathrm{{\scriptscriptstyle
S}}}(\bm{r})\right]\phi_{j}(\bm{r})$
$\displaystyle=\epsilon_{j}\phi_{j}(\bm{r})\;,$ (4)

a set of $N_{e}$ one-particle Schrödinger equations, where
$v^{\tau}_{\mathrm{{\scriptscriptstyle S}}}(\bm{r})$ denotes the Kohn-Sham
potential. The self-consistently calculated Kohn-Sham potential includes both
the electron-ion interaction and the electron-electron interaction, the latter
in terms of a mean-field description. The electronic density is obtained from
the Kohn-Sham orbitals $\phi_{j}(\bm{r})$ as

$\displaystyle n(\bm{r})$ $\displaystyle=\sum_{j}\,|\phi_{j}(\bm{r})|^{2}\;.$
(5)

The total energy functional is expressed within the Kohn-Sham formalism as

$E_{\mathrm{total}}=T_{\mathrm{{\scriptscriptstyle
S}}}[n]+E_{\mathrm{{\scriptscriptstyle H}}}[n]+E_{\mathrm{{\scriptscriptstyle
XC}}}[n]+E^{ei}[n]+E^{ii}\;.$ (6)

In Eq. (6), $E_{\mathrm{{\scriptscriptstyle H}}}[n]$ denotes the Hartree
energy, i.e., the energy contribution from the electrostatic interaction of
the density with itself, $E^{ii}$ the energy contribution from the ion-ion
interaction, and $E^{ei}[n]$ the energy contribution from the electron-ion
interaction. $T_{\mathrm{{\scriptscriptstyle S}}}[n]$ represents the kinetic
energy of the Kohn-Sham system. The final energetic contributions are the
exchange and correlation energies $E_{\mathrm{{\scriptscriptstyle XC}}}[n]$.
Given that the form of $E_{\mathrm{{\scriptscriptstyle XC}}}[n]$ was known, it
is ensured that DFT calculations yield _in principle_ the exact result. In
practice however, $E_{\mathrm{{\scriptscriptstyle XC}}}[n]$ has to be
approximated. The fundamental approximation to $E_{\mathrm{{\scriptscriptstyle
XC}}}[n]$ is the local density approximation (LDA) [22, 23], which by
incorporation of the density gradient or further constraints can be expanded
to (meta/hybrid-) generalized gradient approximations ((meta/hybrid-)GGA,
e.g., PBE [24], SCAN [25], or B3LYP [26, 27]). Notable extensions of DFT are
time-dependent DFT (TD-DFT) [28] (representing the time-dependent Schrödinger
equation in terms of the single-particle picture) and finite-temperature DFT
(FT-DFT), which extends DFT to temperatures $\tau>0$ [29, 30, 31, 32].

###### Beyond DFT

In contrast to DFT, wavefunction methods center on the many-particle
wavefunction $\Psi(\underline{\bm{r}})$. The simplest methods is Hartree-Fock
which constructs a single Slater determinant, but neglects electronic
correlation by construction [33]. Thus, several post-Hartree-Fock methods have
been developed that incorporate electronic correlation explicitly, such as the
coupled-cluster method (CC / CCSD / CCST) [34, 35, 36, 37, 38, 39],
configuration interaction (CI) [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50],
and Møller-Plesset perturbation theory (MP) [51]. These methods generally
outperform DFT calculations in terms of accuracy [52], but do so at a much
higher computational cost.

Electronic structure calculations can also be carried out using Monte-Carlo
methods, i.e., methods that use repeated, randomized sampling, allowing for
generally trivial parallelizability and a formally exact quantum mechanical
treatment of the electronic structure problem. Examples of such methods are
variational Monte Carlo [53], fixed-node diffusion Monte Carlo [54], and path-
integral Monte Carlo [55, 56], useful for $\tau>0$.

###### Coupling DFT to Molecular Dynamics

Investigating the electronic structure of a material is often only a part of a
bigger systematic study. Simulations on larger time and length scales are
often realized by combining classical mechanical molecular dynamics (MD) of
the ions with electronic structure calculations. To this end, one calculates
an interatomic potential (IAP), which determines how the ions in the system
interact. Such IAPs can also be replaced with reasonably fast quantum
mechanical calculations (DFT-MD) or be constructed using electronic structure
data for later use. Traditional IAPs are the Lennard-Jones potential [57], the
embedded atom model (EAM) [58] or other many-body potentials [59, 58, 60, 61,
62, 63, 64, 65, 66]. Recently, ML has given rise to a new class of IAPs that
are constructed using data-driven algorithms (see Sec. II.2). Notable examples
include ANI-1 [67], GAP [68], BLAST [69], HIP-NN [70], SchNet [71], SNAP [72],
DPMD [73], and the AGNI ML force fields [74]. The resulting IAPs come well
within chemical accuracy while retaining an often negligible computational
cost.

###  II.2 Machine learning

ML is a form of computational pattern recognition which has had profound
impact on multiple fields over the past decade. Advances in GPU hardware
coupled with the development of high-level frameworks has led to many
different applications in various fields. These applications include
autonomous driving [75], high-energy particle-physics data analysis [76], and
materials discovery [77]. In this section, a brief background on ML is given,
as well as some specifics relevant to most common models in materials science
applications.

####  II.2.1 Introduction

While there is no standard definition of ML, it is commonly defined as the
study of computer algorithms that improve automatically through experience
[78]. One usual way to divide ML models is into supervised models (models that
make a prediction with respect to a ground truth), unsupervised models (models
that operate on data without an underlying ground truth, and perform, e.g.,
dimensionality reduction) and reinforcement learning (models that explore
complex environments). ML models can be further divided into parametric and
non-parametric models. When the model involves a finite set of learning
parameters $\mathbf{W}$, the model is said to be parametric. These parameters
are not set explicitly and learned iteratively through the learning process.
Models without a finite set of parameters are non-parametric. In non-
parametric models, the set of weights $\mathbf{W}$ is generally infinite.

Some examples of ML models include _supervised parametric models_ (linear
regression, neural networks), _unsupervised parametric models_ (Gaussian
mixture models, generative adversarial networks [79]), _supervised non-
parametric models_ (Gaussian Process Regression (GPR) [80], Decision Trees
(DT) [81], Support Vector Machines (SVM) [82]), and _unsupervised non-
parametric models_ (k-means algorithm) [83].

Learning models can also be combined into ensembles. Using multiple learning
models reduces overfitting and improves generalisability. Examples of ensemble
learning techniques are Random Forests (Regression) (RF(R)) [84] and Gradient
Boosting Regression (GB(R)) [85].

Supervised parametric learning problems generally follow a common structure.
Given a dataset $D$ consisting of $N$ samples $\mathbf{X}$ and labels
$\mathbf{y}$, a model $M:\mathbf{X}\mapsto\bm{y}$ with parameters $W$ is to be
fitted such that the cost or loss function $J$ is minimised. For this fitting
only a subset of the dataset is used (training data), and once the model is
sufficiently trained, its performance is analyzed on a separate subset of data
(testing data). If the model performance is satisfactory, it can be used to
predict $\bm{y}$ on input data for which it is unknown (inference). Care has
to be taken during training so as to avoid over- or underfitting a model,
i.e., introducing a high error on unseen data or not capturing the complexity
of the underlying function, respectively. The optimization of the model is
often done by gradient descent algorithms [86] in an iterative fashion.
Lastly, one has to consider so called hyperparameters, i.e., parameters that
describe the model design and are set before model optimization (e.g.,
characterization of the training process, kernel or activation functions).
Their choice is crucial for reasonable model performance.

####  II.2.2 Supervised Learning Models

A brief description of commonly used supervised learning models in materials
science is given in the following.

###### Linear regression

Linear regression [87] is one of the most used algorithms for regression
tasks. Given a dataset ($\mathbf{X},\bm{y}$) consisting of labeled data where
$(\bm{x}^{(i)},{y}^{*(i)}),\leavevmode\nobreak\ i=1,...N$ with
$\bm{x}^{(i)}\in\mathbf{X}$ and ${y^{*}}^{(i)}\in\bm{y}$, regression is used
to determine the relationship between dependent variables $y^{*(i)}$ and
independent variables
$\bm{x}^{(i)}=\\{{x^{(i)}_{1}},{x^{(i)}_{2}},{x^{(i)}_{3}},...,{x^{(i)}_{d}}\\}$,
where $d$ is the dimension of the data. Given an input $\bm{x}^{(i)}$, the
prediction $y^{(i)}$ is defined as

$y^{(i)}=\mathbf{W}^{T}\bm{x}^{(i)}\,,$ (7)

where $\mathbf{W}$ are the learnt parameters of the model, determined by
minimizing a least squares cost function. While there exists an analytical
solution for the optimal value of parameters $\mathbf{W}^{*}$, the
computational complexity for calculating $\mathbf{W}^{*}$ is
$\mathcal{O}(N^{3})$ which becomes intractable for large data sets with high
dimensions. If the number and range of the parameters is ill-suited for the
dataset, the model might reach a sub-optimal state due to overfitting or
underfitting. To overcome this, a regularisation term is added to the cost
function; depending on the choice of this cost function one arrives at methods
such as the Least Absolute Shrinkage and Selection Operator (LASSO) [88] or
ridge regression [89]. Sure Independence Screening and Sparsifying Operators
(SISSO) [90] is an extension to LASSO with better performance for feature
selection. Kernel Ridge Regression (KRR) [91] is a non-parametric realisation
of ridge regression. Instead of learned parameters $\mathbf{W}$, a kernel
function measuring the distance between different data points is used. Linear
regression models are not well suited for classification problems. When
classification is needed, logistic regression [92] can be applied. Here, the
input data is mapped to discrete labels by applying a sigmoid function on the
linear regression model.

###### Neural networks

Neural networks (NNs) are a promising approach to regression due to the
_universal approximation theorem_ [93] that shows that any function can be
modeled by a sufficiently sophisticated NN. NNs have been widely applied to
problems in different domains including image classification [94] or
biomedical engineering [95]. However, neural networks also have some
challenges in determining their hyperparameters and training routines that
avoid overfitting.

##### Introduction:

A neural network is essentially a set of nested linear regression functions
with non-linear activation functions. The base unit of a neural network is a
perceptron or artificial neuron [96]. For data point $\bm{x}^{(i)}$, the
perceptron is defined as

$f^{(i)}=\sigma(\mathbf{W}\bm{x}^{(i)}+b)\,,$ (8)

where $\sigma$ is a non-linear activation function, $\mathbf{W}$ consists of
the weights, and $b$ denotes the bias. When nested together in layers, the
model is known as multilayer perceptron or NN. The output of each layer is
input to the subsequent layer. The first layer, that accepts the input
samples, is called the input layer, and the final layer is called the output
layer. The layers in between are referred to as hidden layers. A layer is
defined by the number of its neurons which is often denoted as width. A NN is
defined by the number and structure of hidden layers. A network with a large
number of hidden layers is called a deep neural network (DNN). This gives rise
to the term deep learning [97]. The design of a NN depends on the learning
task. To succesfully employ NNs, one has to make choices as to their
activation functions [98] (e.g., Tanh and ReLU [99]), loss function and
optimization scheme (standard gradient descent can be replaced by stochastic
gradient descent [100] or the ADAM algorithm [101]).

##### Architectures:

The operations performed by the neurons can be configured for each layer. The
following are some of the most common architectures:

  * _Fully Connected Neural Network (FCN) / Feed-Forward Neural networks:_ Each neuron in a layer of FCNs is connected to every neuron in the next layer [102].

  * _Convolutional Neural Network (CNN):_ CNNs are useful for tasks with 2D or 3D data such as image classification [103].

  * _Graph Neural Network (GNN):_ Instead of a vector input, GNNs [104] can be used with data points in the form of arbitrary graph structures, useful for tasks like many-body problems in cosmology [105].

  * _Recurrent Neural Network (RNN):_ RNNs are used for tasks that involve sequential or temporal evolution such as language translation [106].

  * _Physics Informed Neural Network (PINN):_ PINNs are used to solve partial differential equations. This is done by incorporating initial and boundary conditions in the loss term [107].

NNs are effective at modelling complex non-linear functions and can be
generalised well to new data [108]. Shortcomings of NNs are that large
training datasets are required for practically useful results, and that the
black-box models lack explainability and interpretability [109].

###### Gaussian Process Regression

Gaussian Processes (GPs) or Gaussian Process Regression (GPR) is a powerful
non-parametric supervised learning technique with robust uncertainty
quantification. It can be thought of as a distribution over functions. By
constraining the probability distribution at training points, a learned
function can be drawn over the test points. Instead of learning any learning
parameters $\mathbf{W}$, inference is done by constructing the appropriate
covariance matrix and drawing functions from it. Such a covariance matrix is
defined by using a kernel function $k$, the choice of which impacts the
usefulness of the model. With appropriate kernels, GPs are equivalent to other
ML methods such as neural networks [110]. GPs provide in-built uncertainty
quantification and are effective for small datasets. However, the inference
step involves matrix inversion which scales as $\mathcal{O}(N^{3})$. This
makes GPs computationally expensive for larger datasets. Approximate inference
to increase the scalability of GPs is an active area of research [111].

##  III Machine-learning in Materials Science and Chemistry

Electronic structure methods (Sec. II.1) suffer from their unfavorable
computational scaling, as even the most efficient DFT implementations enable
calculations with only few thousand atoms [112]. Increasing computational
power can only alleviate this problem to a certain degree. Data-driven
workflows (Sec. II.2) offer speedups by augmenting or replacing standard DFT
calculations.

Figure 2: Comprehensive map of workflows combining ML and DFT in current
practice. The map is color-coded: the colors highlight the five main
categories we have identified in our comprehensive literature survey; the
colors also correspond to those used in Fig. 3 which illustrate the results of
our citation analysis. More specifically, blue corresponds to Sec. III.1,
orange to Sec. III.2, green to Sec. III.3, red to Sec. III.4, and purple to
Sec. III.5.

In the following, we provide an overview over the most recent research
developments combining ML and DFT aimed at materials science and chemistry
applications. We have grouped them into five main categories based on the type
of approach they employ and we highlight the most representative articles.
With this grouping, we identify the overall ML-DFT workflow visualized in Fig.
2. By means of this diagram we link the individual categories presented in the
following.

###  III.1 Property mappings

Perhaps the most intuitive approach to employing ML techniques in DFT is by
learning distinct physical or chemical properties from data sets. While
accessing these through DFT calculations is computationally cheap when
compared to other methods of comparable accuracy, the computational cost can
often be restrictive when investigating large systems. Likewise, such
investigations are often focused on a specific property relevant to a given
application rather than on a fundamental understanding of the electronic
structure. Hence a feasible approach is to train ML models to replicate a
mapping of the form $\underline{\bm{R}}\rightarrow p$ for a relevant property
$p$ that one would initially obtain from a DFT calculation or from post-
processing. In the following, an overview of such approaches is given, grouped
by their respective application domain. Noteworthy high-impact results are
discussed in detail.

####  III.1.1 Chemical Reactions

The demand to both reduce and capture carbon emissions motivates improving the
efficiency of chemical reactions employed in industrial settings and searching
for novel reactive pathways. In the context of computational materials science
and chemistry this usually translates to identifying catalysts for reactions
of interest (for instance $\mathrm{CO}_{2}$ reduction) and their respective
activity and selectivity. The computational demand of such studies can be
drastically decreased by employing ML, and a wide range of such investigations
has been performed [113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123,
124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138,
139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153,
154, 155, 156, 157, 158, 159, 160].

A typical target quantity is the adsorption energy of a molecule on a
catalyst. In Ref. [128], the $\mathrm{CO}_{2}$ reduction is analyzed where its
efficiency is highly dependent on the adsorption properties of CO and H on the
chosen catalyst. The catalyst was a high entropy alloy (HEA) consisting of at
least five elements [161]. The configuration space of possible potential
surface adsorption sites is vast. A GPR model built on a subset of those
enables the prediction of adsorption energies on the entire configuration
space. In this manner, a candidate catalyst for this reaction was identified,
which was then also studied experimentally [162]. Similarly, Ref. [122] also
uses GPR to predict activation barriers of dihydrogen on metal complexes.
There are various approaches of using ML models to facilitate the design or
discovery of catalysts by performing such high-throughput searches. Other
notable examples include the $\mathrm{N}_{2}$ electroreduction on transition
metal alloys using GNNs [130] and the combination of linear models based on
experimental data and RFR/GBR models on ab-initio data to directly predict
activities and selectivities for ethanol reforming on bi-atom catalysts[117].

Ref. [114], which is concerned with the $\mathrm{O}_{2}$ reduction on bi-atom
catalysts, follows a slightly different approach. Here, ML is used to unveil
correlations in the underlying DFT data. By training a RF regressor to
correctly predict adsorption energies from a number of chemical descriptors,
the correlation between the adsorption energy and these descriptors is
evaluated. Using ML to identify the relative importance of individual
descriptors, i.e., catalyst design features, is a technique used for a variety
of reaction/catalyst combinations. These include $\mathrm{CO}_{2}$ capture on
MOFs [139], $\mathrm{O}_{2}$ reduction on single atom catalysts [134], olefin
oligomerization on Cr based catalysts [154], and the hydrogen evolution
reaction (HER) [141].

####  III.1.2 Discovery of Novel, Stable Materials

The discovery of novel materials is at the very core of computational
materials science, as computational methods reduce the cost of exploring large
configuration spaces drastically compared to experimental methods. The ever
increasing complexity and size of these configuration spaces makes their
exploration an ideal application for ML methods that can be employed to
accelerate such investigations [163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196].

Such studies usually sample a certain space of possible candidate structures
using some measure of stability, and possibly, specific values for properties
of interest. Ref. [177] uses DNNs and a minimal set of chemically motivated
descriptors (electronegativity and ionic radii) to analyze a range of mixed-
series crystals and perovskites. The formation energy is then predicted for
these materials within chemical accuracy but at negligible computational cost.
This enables rapid identification of stable crystal structures which can then
be synthesized in the lab. A similar approach is employed in Ref. [188], where
a class of multicomponent Ti-alloys are investigated. It extends the previous
study by prediciting elastic properties, which are important for aerospace
applications. The combination of stability metrics with application-centered
properties is also demonstrated in Ref. [189], where a SISSO model is trained
to predict formation enthalpies of 2D materials to assess their thermodynamic
stability. The proposed application domain of this method is the
identification of materials suitable for photoelectrocatalytic water
splitting. Here, the band gap is the primary metric. While the presented model
is not capable of predicting the band gap, it significantly speeds up the
search for structures by reducing the search space by more than a half. A
similar example of screening a chemical for thermodynamic stability is Ref.
[170], where hybrid organic-inorganic perovskites (HOIP) for photovoltaics are
at the center of attention.

A somewhat unconventional approach is followed in Ref. [176] where a CNN is
used. The CNN is directly applied to the periodic system of elements. To this
end, a given structural blueprint is represented as a matrix that takes the
form of the periodic table (periodic table representation, PTR). The entries
of that matrix reflect the proportion of the elements present in the material.
The CNN then predicts formation energies and lattice parameters from these
input matrices. It is important to note that these input matrices not only
carry information on the proportionalities of the involved elements as an
arbitrary list would, but also information on chemical relationships in the
form of positions of elements in the periodic table.

####  III.1.3 Electric Properties and Photovoltaics

Driven by the effort towards a sustainable future, another important
application domain is the search for materials useful in electronic
applications such as batteries for electric cars, efficient photovoltaics, and
semiconductors for microelectronics. While the underlying problem is similar
to before, i.e., finding suitable materials from large configuration spaces,
here the constraints on suitability are based on electrical properties such as
the band gap or the photoelectric conversion coefficient (PCE). ML has been
employed in a number of studies [197, 198, 199, 200, 201, 202, 203, 204, 205,
206, 207, 208, 209, 210, 211, 193, 212, 213, 214, 215, 216, 217, 218, 219,
220, 221, 222, 223, 224].

A lot of studies are directed towards photovoltaic materials, for which the
PCE, i.e., the percentage of light transferred to electrical energy by a
photovoltaic material, is one of the most important metrics. In Ref. [198],
Bayesian-Regularized NNs were trained on DFT data and used to predict the PCE
for organic photovoltaic materials (OPVs). A prediction accuracy of $\pm
0.5\%$ was achieved, allowing for efficient screening of promising materials.
The basis for these models are simple, chemical descriptors. A similar
strategy has been employed in Ref. [208], where simple structural and
electronic similarity features served as inputs for KRR models, again for
OPVs. The investigation of photovoltaic materials is not limited to PCE
models.

Another important target quantity for ML approaches is the band gap, as
demonstrated in Ref. [197], where the vast chemical space of HOIP is sampled,
similar to the aforementioned Ref. [170], however with an emphasis on the
electrical properties of candidate materials. Using a small set chemical
descriptors band gaps computed using DFT were learned and predicted by GBR.
The results of Ref. [170] were incorporated in this investigation by means of
progressive ML [225], in order to enrich the underlying database of the ML
model. Yet, such models suffer from the underestimation of band gaps by usual
DFT functionals, an issue addressed in Ref. [212]. The established way of
overcoming these inaccuracies inherent to DFT are by using many-body theories,
such as the GW approximation, which are computationally very costly. By using
GPR models, the corrections applied by the GW approximation [226, 227, 228] to
band gaps obtained from DFT can be evaluated in a fraction of the time and
independently from DFT, since the GPR models are only directly dependent on
chemical descriptors.

Other electrical properties of interest addressed in ML-based studies are
piezoelectric and dielectric responses of inorganic materials [223], Li ion
conductivity for Li ion batteries [209] or impurity levels of semiconductors
[202].

####  III.1.4 Elastic and Structural Properties

The design and discovery of materials with specified structural properties
constitutes another important research thrust where DFT data and property
mappings in terms of ML play a central role. These investigations [229, 230,
231, 232, 233, 234, 235, 236, 237, 238, 239, 240] are similar in nature to the
ones discussed in Sec. III.1.2, however, with a stronger emphasis on property
selection.

Typical quantities of interest in this context are elastic constants or
Young’s modulus. NNs and RFRs were used in Ref. [229] to predict bulk, shear,
and Young’s modulus for ternary Ti-Nb-Zr alloys in order to accelerate an
otherwise costly high-throughput DFT search for low-modulus alloys, similar to
Ref. [188]. Identification of such alloys is in demand from the biomedical
industry, and due to the ML assisted study, a promising candidate could be
found.

Another important property often considered in high-throughput calculations is
the elastic constant. This is addressed by Ref. [236] and [238] in terms of NN
models to provide both direct predictions of elastic constants.

ML models have been also used to unveil relationships between structure and
properties of materials, such as in Ref. [234] by means of a GPR based
analysis of molecular crystals. Here, similarities between crystal structures
were classified along with the prediction of lattice energies where the
resulting structural landscapes shed light on molecular crystallization.

####  III.1.5 Quantum Chemical Information

While there is no precise distinction between quantum chemistry and materials
science, these closely related fields can by distinguished in terms of either
the systems considered or the properties under investigation. While the
articles and groupings of property mappings discussed above are drawn from
both quantum chemical and materials science applications, ML approaches are
also heavily utilized for the rapid prediction of quantum chemical properties,
such as polarizabilities, partial charges, HOMO-LUMO levels or bond
dissociation energies [241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251,
252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266,
267, 268, 269, 270, 271, 272, 273].

In Ref. [257], polarizabilities are calculated with both DFT and CCSD. The
latter provides better overall accuracy at a drastically increased
computational cost. GPR models trained on either of the resulting data sets
are capable of predicting polarizabilities accurately. Yet the best
performance with respect to the baseline provided by CCSD is achieved when
learning and predicting the difference $\Delta$ between these two levels of
theory and then applying the output of such a model to DFT results. Such an
approach has been followed in combined ML/DFT applications (see Sec. III.3 and
the approaches for exchange-correlation functionals discussed therein) and is
often referred to as $\Delta$-learning [274]. It was originally introduced for
the prediction of energies. Another example in the context of these target
quantities is Ref. [246], where NNs predict corrections on top of DFT-level
energies for non-covalent interactions.

Other approaches follow an outline more consistent with the workflows
described above, such as Ref. [245], wherein a GNN is utilized to predict bond
dissociation energies for a large database of organic molecules. The resulting
model can be applied for instance in drug design. Similarly, in Ref. [265] NNs
are used to predict spin-state splittings of transition metal complexes,
employing a test data set that includes experimental data in addition to DFT
data.

Related to these works are applications of TDDFT, such as Ref. [251], wherein
corrections to DFT calculated electronic spectra are learned from CCSD
calculations within the $\Delta$-learning methodology.

Finally, another interesting approach is presented in Refs. [242] and [271].
Models based on KRR and NN were employed to predict whether a specific
configuration should be treated with DFT or more accurate (and costly)
multireference methods.

####  III.1.6 Magnetic, Thermal, and Energetic Properties

Finally, there are ML property mappings that do not directly fall into the
application-based groupings above. Properties that can be derived from DFT and
predicted or analyzed with ML methods but have not yet been discussed at
length are, for instance, transport and diffusion properties [275, 276, 277],
surface energies [278], and magnetic properties [279].

Some studies that combine ML and DFT fall into scientific disciplines not
mentioned yet, such as studying RNA conformations [280] or the DFT prediction
of nuclear magnetic resonance shifts (NMR) [281, 282, 283]. DFT can also be
used as a pre-processing or pre-screening tool for ML workflows based on
experimental data, as demonstrated in Ref. [284] and Ref. [285], where Curie
and transition temperatures are learned, respectively. A purist’s approach to
combining DFT and ML is taken in Ref. [286], which is concerned with melting
temperatures.

Lastly, an important direction of research is concerned with energetic
materials, i.e., materials that hold large amounts of chemical energy, such as
fuels or explosives. A comprehensive study of ML techniques for energetic
materials can be found in Ref. [287].

###  III.2 Interatomic Potentials

A specialized type of property mappings in terms of ML are interatomic
potentials (IAPs) (Sec. II.1.2). Here, a ML mapping of atomic positions to the
potential energy and atomic forces is performed, also referred to as force
field or potential energy surface (PES). Most commonly, these ML-IAPs
parametrize the Born-Oppenheimer PES defined in Eq. (3). The trained models
can be used to perform MD simulations at an accuracy close to the first-
principles methods used for the calculation of the training data (e.g., DFT)
with no significant computational overhead. In the following, an overview of
developments in the field of ML-IAPs is given The reviewed publications have
been grouped by the type of IAP used.

####  III.2.1 Neural Network Potentials

A large number of IAP workflows rely on neural networks in order to
approximate energies and forces from atomic positions [288, 289, 290, 291,
292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306,
307, 308, 309, 310, 311, 273, 312, 313, 314, 315, 316, 317, 318, 319, 320,
321, 322, 323, 324, 325].

The type, architecture, and data representation of the employed NNs vary
drastically between different approaches. Arguably the most common type of NNs
within these approaches are deep feed-forward networks as used in Ref. [291].
Using such _deep potentials_ , the authors were able to investigate a proposed
liquid-liquid state transition in water, a study unattainable with regular DFT
calculations. Similarly, in Ref. [304] an investigation of the phase-change
material GeTe, which is highly relevant for the development of non-volatile
memory devices, is presented. These simulations would have been challenging
with classical IAPs. Instead, a NN potential is used, drawing on Ref. [320]
and Ref. [321]. The transferability of models highlights another advantage of
ML-IAPs in comparison to highly specialized workflows, as they aim to replace
DFT or classical potentials with often no application-induced constraints.

Traditional NNs can be improved by incorporating physical considerations, such
as in Ref. [309]. The thought behind developing such physically enhanced
networks is to enable extrapolation capabilities outside of the range of
training data by encoding information on the chemical bonding of the atomic
structures. Extrapolation is traditionally a problematic task for NNs, yet
often necessary when performing simulations on new materials. They demonstrate
the improved capabilities of such IAPs based on physically enhanced NNs for
the case of aluminum.

The combination of DFT and ML is also leveraged to construct IAPs with an
accuracy better than DFT itself, as shown in Ref. [307]. There, transfer
learning is used to construct an IAP with CCSD accuracy. First an IAP is
calculated using DFT data which is retrained via transfer learning on CCSD
data. This enables rapid predictions of total energies with CCSD accuracy. A
similar approach is taken in Ref. [326]. Here, a neural network is trained to
correct a classical IAP to within DFT accuracy by training it on DFT data.
Both of these approaches follow the principal idea of $\Delta$-learning
mentioned in Sec. III.1.5.

####  III.2.2 Gaussian Approximation Potentials

Another important class of IAPs are Gaussian Approximation Potentials (GAPs),
primarily based on Ref. [68]. Based upon GPs, the total energy is decomposed
into atomic contributions, which are then learned by GPR from DFT data.
Information on the atomic neighborhoods is encoded by a bispectrum
representation of the atomic density. A range of publications have employed
GAPs to model the PES [327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337,
338, 339, 340, 341, 342, 343].

For example, in Ref. [342], a GAP is trained for amorphous carbon. This
potential is then used by the same authors in Ref. [339] to investigate
electrode materials based on amorphous carbon through a combination of GAP and
MD, a task usually unattainable with quantum mechanical accuracy. The same GAP
is also employed in Ref. [343] and [341], further highlighting the universal
re-usability of these IAPs.

A GAP has also been used to model phase-change materials [338], similar to
Ref. [304] which was based on NNs. As one would expect, there exists no
singular optimal choice for IAPs, and several different ML workflows can be
used to tackle similar problems. Furthermore, GAPs have been shown to be able
to model complicated PES of magnetic materials in Ref. [340].

Finally, GAPs can be incorporated in larger ML workflows, such as the CALYPSO
structure prediction method [337]. They also facilitate automatic workflows
for optimal IAP construction [344].

####  III.2.3 Moment Tensor Potentials

Similar to GAPs, moment tensor potentials (MTPs) [345] are based on linear
regression and polynomial expansion. They can, in principle, approximate any
PES. MTPs pose another class of powerful ML-IAPs that have been in used in
many publications [346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357,
358, 359, 360, 361, 362, 363, 364, 365], often in conjunction with active
learning approaches.

For example, Ref. [349] investigates solid-state batteries, which are of high
interest due to the increasing demand for powerful and reliable electrical
power sources. They employed a MTP that is trained on the fly that improves
the accuracy of the IAP with ab-initio data where needed.

Using MTPs for active learning was first proposed by the authors of Ref.
[345]. A linearly parameterized IAP is constructed on the fly based on the
D-optimality criterion [350]. This method has subsequently been used to study
diffusion processes [355] or lattice dynamics [354]. As the authors point out,
this technique is not limited to MTPs and can be employed for other linear
IAPs as well.

Another prominent application of MTPs is the investigation of both phononic
properties and thermal conductivities [346, 362, 347, 365] and the search for
new alloys [356].

####  III.2.4 Spectral Neighborhood Analysis and Other Potentials

Given the amount of possibilities ML-DFT techniques provide, it is no surprise
that there are a number of IAP construction methods [366, 367, 368, 322, 369,
325, 324, 370, 371, 372, 344, 373, 160, 374, 375, 323, 72, 376] that do not
fall into the three categories outlined above.

The most prominent of these is the spectral neighborhood analysis potential
(SNAP) [72, 377, 378, 376]. It is based on a linear model, similar in
construction to GAP and MTPs. SNAP has been used to investigate lithium
nitride [379], carbon under extreme conditions [380], and molybdenum alloys
[367, 373].

Other IAP approaches draw on a range of different ML techniques, such as GPR
or KRR [366, 375, 381, 372, 368, 370, 369]. Of note is further the (s)GDML
approach [382, 383, 384, 385, 386], which combines a kernel based ML approach
(GDML) with physical symmetries (sGDML) to construct efficient and accurate
IAPs.

Finally, another procedure for constructing ML-IAPs is based on optimizing
classical IAPs. This is done in Ref. [374], where a genetic algorithm is
utilized to fit a bond order potential to investigate metal-organic
structures.

####  III.2.5 Technical Aspects for the Constructing of Potentials

Finally, there are technical aspects when dealing with ML-IAPs. The
construction of these IAPs is often a recurring task, since one is often not
only interested in a specific system at specific conditions, but in rather
large-scale systematic studies. The methodologies outlined above are not
limited to specific elements or conditions apart from limitations introduced
by DFT itself. Therefore, frameworks exist to simplify the repeated
construction of such potentials. These include PANNA [387], ANI-1 [388],
TensorMol [389] and Amp [390], which provide frameworks to build IAPs based on
NNs. For MTPs, MLIP [391] provides a software framework incorporating active
learning. GAPs can be build using the QUIP code [392] and SNAPs can be
constructed with the LAMMPS [393] code.

Two other closely related aspects worth mentioning in this context are active
learning and uncertainty quantification. Ref. [394] introduces a framework to
realize such active learning procedures for IAPs, while Ref. [395] introduces
potentials based on dropout-uncertainty neural networks (DUNN) which are
capable of assessing uncertainties in their predictions. Such estimations are
important, because NNs usually do not perform well during extrapolation tasks,
yet such situations might occur during automated materials calculations.

###  III.3 Learning Electronic Structures

While the preceding sections discussed ML approaches which center on specific
properties extracted from DFT data, be it a wide variety of materials
properties (Sec. III.1) or specifically the energy-force-landscape (Sec.
III.2), the emphasis in the following is on using ML techniques to tackle the
electronic structure problem directly. In these investigations, DFT is not
solely a data acquisition tool, but rather the centerpiece of a ML-powered
workflow aimed at a better understanding of electronic structures.

The most common mapping deals with the electronic exchange-correlation energy
[396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410,
411, 412, 413, 414, 415, 416, 417]. As discussed in Sec. II.1.2, the accuracy
of the chosen exchange-correlation approximation primarily determines the
accuracy of the overall DFT simulation. The exchange-correlation functionals
discussed in Sec. II.1.2 mainly employ exact conditions for their
constructions, but fitting to either experimental or high-level calculations
data is also a common practice for the construction of exchange-correlation
functionals. These have been in use for a long time (see Ref. [418]),
especially in quantum chemistry. Using ML methods to perform similar fits is a
natural consequence. An example of this is Ref. [401]. Here, the exact
exchange-correlation energy is calculated for a system of limited size and
then approximated using a NN, where the electronic density serves as the input
quantity. By using automatic differentiation, the exchange-correlation
potential is obtained as well. The resulting model incorporates non-local
effects, the lack thereof being a usual shortcoming of traditional exchange-
correlation functionals at a negligible computational overhead compared to
exact calculations. A similar approach is taken in Ref. [407], which
introduces NeuralXC, a framework for constructing such NN exchange-correlation
functionals, or Ref. [397], where CNNs are used to estimate the exchange-
correlation energy from the density by drawing on the convolutions of the CNN
architecture to featurize the electronic density. They recover the exact
exchange of the B3LYP functional. This is especially important given that the
B3LYP functional achieves exact exchange by evaluating the Kohn-Sham orbitals
rather then the density, thus demonstrating how NNs can extract meaningful
information from the electronic density. Improving exchange-correlation
energies with ML is not necessarily limited to the construction of ML
functionals. In Ref. [396], range-separation parameters for the established
long-range corrected Becke-Lee-Yang-Parr (LC-BLYP) functional [27, 418, 419,
420, 421, 422] are learned from molecular features. The application of exact
conditions in combination with ML functionals is also a subject of current
research. As shown by Ref. [415], the kinetic energy functional rather then
the exchange-correlation functional is investigated.

Instead of directly learning the exchange-correlation functional itself,
corrections to these functionals can be learned. This $\Delta-\mathrm{DFT}$
method was already briefly introduced in Sec. III.1.5 and has been applied in
Ref. [404]. Here, differences between DFT and CC predicted total energies were
learned using a KRR model. This model is in turn used to enhance DFT
calculations towards chemical accuracy at almost no computational overhead,
thereby, enabling highly accurate MD simulations. The advantage of methods is
that constructing an ML correction rather then a ML functional reduces the
training samples drastically. This is also shown in Ref. [410], where non-
covalent interaction corrections to standard DFT functionals are learned at a
much smaller cost.

Going beyond the electronic exchange-correlation energy, other studies focus
on intermediate electronic quantities. In Ref. [423], pattern-learning
techniques are used to predict the density of states (DOS) of an alloy system
based on chemical features with high accuracy. The DOS is an important
quantity in solid-state physics that can be used to discover or assess the
properties of new materials in terms of their band energy. Ref. [424] utilizes
a similar workflow to predict the DOS, but further includes uncertainty
estimates. This workflow is mainly applied to silicon structures. The authors
further compare the results of deriving the band energy from a predicted DOS
with ML models that directly predict the band energy. They find that the
former approach outperforms the latter. This makes intuitive sense given that
the ML model learns from a larger amount of data with direct access to
information about the electronic structure.

Another logical quantity of interest in this context is the electronic density
itself, for which various ML approaches exist [425, 426, 427, 428, 429]. Most
notably, Ref. [425] uses KRR to learn and predict electronic densities,
although these densities are in turn only intermediate quantities. The
principle quantity of interest is the total energy. Similar to the results of
Ref. [424], it is found that learning the electronic density from atomic
positions and then the total energy from the electronic density, leads to more
accurate results than can be achieved by a direct mapping of atomic positions
to total energies. It should be noted that the mapping from the electronic
density to the total energy has to be done via ML in the context of KS-DFT,
since the KS energy functional includes terms that are only implicit
functionals of the density, namely the kinetic energy of the non-interacting
system, which depend on the KS wave functions explicitly.

In order to gain direct access to the total energy, another target quantity is
needed. One possible candidate is the local density of states (LDOS), which
gives the DOS at any point in space. In Ref. [430], NNs are used to predict
both the electronic density and the LDOS individually at each grid point. By
integrating the LDOS over the real space afterwards, the DOS is obtained. We
have recently proposed a similar ML workflow [431] that follows along similar
lines. It uses predicted LDOS values to obtain the total energy from the
infered DOS and density. Predicted densities can also be used to construct a
$\Delta-$DFT [432] approach. On the other hand, Ref. [433] and Ref. [434] are
concerned with the ML-aided construction of density functionals that enable an
efficient prediction of the total energy directly from electronic densities,
irrespective of how these densities are actually obtained.

However, electronic structure theory is not limited to an energy-based
analysis. Refs. [435] and [436] apply ML methods to Hamiltonians in order to
calculate different quantities of interest. Similarly, Ref. [437] uses GPR to
fit potentials on DFT data, that can be used to improve calculations at lower
levels of theory (e.g., density functional tight-binding). Finally, ML can
also be used to accelerate the numerical treatment of DFT by reducing the
numerical overhead of the solution of the KS equations [438] or identifying
efficient, adaptive basis sets [439].

###  III.4 Other Approaches

There are approaches that do not fit in the categories outlined above, either
dealing with more technical aspects on combining ML and DFT (see Sec. III.5)
or simply mappings and calculations not discussed yet [440, 441, 442, 443,
444, 445, 446, 447, 371] [448, 449, 450, 451].

An example of such an approach is Ref. [441], which is concerned with the
already discussed prediction of NMR shifts, but follows a different scheme to
directly machine-learn chemical shifts from DFT by using DFT calculated
properties as descriptors for a NN. Similar approaches are developed in Ref.
[447] and Ref. [445].

Contrarily, Ref. [444] uses DFT as part of a loss function by calculating NMR
spectra via DFT for molecules generated by a ML algorithm. Differences in DFT
calculated and experimentally measured spectra are used to train this
molecular generator. A similar approach is employed in Ref. [449] for finding
organic molecules. Generally, models trained on data are used as key
components of such search algorithms in the form of an "evaluator" of
candidate structures proposed by a genetic algorithm [448, 450], an
evolutionary algorithm [440], or a Monte Carlo simulation [442].

ML techniques are also used to predict "hyperparameters" of ab-initio
simulations. These are usually adjusted manually through experience or by
converging them with subsequently more costly and accurate calculations. An
example is Ref. [446], where a DT was used to predict the number of k-points
and the cutoff energy in plane-wave DFT calculations. Similarly, in Ref. [443]
the choice of the exchange-correlation functional is predicted from chemical
descriptors. Finally, DFT-based models are also integrated into elaborate
statistical mechanics frameworks as in Ref. [451].

###  III.5 Technical Aspects

Building complicated workflows involving DFT and ML poses several technical
problems that have to be addressed. First and foremost, any data-driven
approach is fundamentally reliant on data. In the context of ab-initio
simulations, data is both costly to acquire and digitally exchangeable by
nature. To that end, it makes intuitive sense to recycle already pre-
calculated data and models in terms of databases, a number of which are given
in Tab. 1, alongside a short overview over useful software frameworks for ML-
DFT approaches. For example, JARVIS includes properties not only on around
40,000 materials, but also ready-to-use ML models trained on DFT data. The
database is constantly expanded and, thereby, facilitates the construction of
new workflows. Similarly, acquiring new training data is facilitated with
integrated workflows [171].

Another important aspect concerns the choices with respect to the ML
methodologies. A wealth of ML techniques can be applied to DFT calculations
and data. Likewise, there is a sizable amount of procedures to encode
information in terms of descriptors. Investigations [452, 453, 454] as to
which of these techniques hold advantages over others are crucial. In Ref.
[454] different types of neural networks (CNN, DNN, single-layer NN) have been
assessed in terms of predicting molecular properties, with a shallow DNN
performing best. Similarly, in Ref. [452], different ways to encode
compositional information for a GB model were compared. None of the
investigated methods was able to produce models capable of assessing the
stability of new materials reliably.

Indeed, the development of new descriptors and techniques to represent
chemical information for specialized use cases is subject of ongoing research
[455, 456, 457, 458, 459, 460, 461, 462, 463, 464]. For example, Ref. [464]
introduces persistent image descriptors (PI), based on concepts from applied
mathematics, to represent chemical structures in the search for functional
groups capable of capturing $\mathrm{CO}_{2}$. Different ML models were
trained on data encoded by these descriptors and compared to other commonly
used descriptors.
Common descriptors are SOAP [465], the Coulomb matrix [466], BoB [467], FCHL
[468], and ACE [469, 470]. This list is not comprehensive, and a number of ML
workflows use customized descriptors or simply easily accessible chemical
information to represent materials. The utility of these descriptors is
demonstrated in the aforementioned publications. Ref. [466, 468, 467] use
their respective descriptors and KRR to learn and predict atomization
energies. On the other hand, Ref. [465] uses SOAP descriptors to construct and
compare GAPs, with Ref. [469] building ACE based IAPs as well.

Lastly, transferability and uncertainty quantitifcation have to be taken into
account in order to apply combined ML-DFT workflows in future applications.
Correctly treating such aspects reduces the need to retrain models. As Ref.
[471] suggests, the transferability of ML models can be utilized with certain
limitations, as demonstrated for moderate volume changes in the liquid phase.
Uncertainty quantification, as investigated in [472], deals with assessing
whether a particular prediction can be trusted or additional data is needed to
improve the model.

Table 1: Non-comprehensive selection of useful software frameworks for constructing ML-DFT workflows. The list of DFT software packages is ordered by the number of citations [473]. Task | Software
---|---
DFT calculations (periodic systems) | VASP [474, 475, 476], QuantumESPRESSO [477, 478, 479], WIEN2k [480, 481], CASTEP [482]
DFT calculations (isolated systems) | Gaussian [483], GAMESS [484, 485], ORCA [486, 487], MOLPRO [488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499]
Neural Network construction | PyTorch [500, 501], TensorFlow [502], Keras [503], Flux.jl [504, 505]
General ML library | Shogun [506], Scikit-Learn [507], Weka [508]
Gaussian Process Regression | GPytorch [509]
Constructing NN-IAPs | PANNA [387], ANI-1 [388], TensorMol [389], Amp [390]
Constructing other IAPs | MLIP [391], QUIP code [392], LAMMPS [393]
ML-exchange-correlation | NeuralXC [407]
Toolchaining | ASE [510], AiiDA [511, 512], DeepChem [513]
Databases | MoleculeNet [514], OQMD [515], ANI-1 [67], PubChemQC [516], NOMAD [517],
| JARVIS [518]

##  IV Discussion

###  IV.1 Citation Analysis

Figure 3: Number of publications and average number of citations per year by
year of publication and category. Average number of citations marks how often
a publication was cited per year after publication (including year of
publication). The total number of publications for which data is shown here
was 350, with data for 15 articles published prior to 2013 not shown for
better visibility. Figure 4: Number of publications by year for property
mapping and interatomic potential related publications.

We conclude this review by assessing general trends in the field in terms of a
citation analysis based on the 365 research articles collected. The first part
of this collection was gathered at the end of 2020 using Web of Science [519].
Grouping of these articles was performed manually. Citation numbers where
retrieved via Google Scholar [520] by an automated python script. Both the
script and the database of research articles are provided alongside this
publication [521].

The results of this citation analysis are shown in Fig. 3. We note that we use
the average number of citations as a measure in Fig. 3. It denotes how often a
research article published in a particular year was cited per year on average
in subsequent years. In this manner, articles published in later years can be
judged on the same basis.

It is apparent from Fig. 3 that the number of publications increases rapidly,
highlighting the growing importance of this research area. The number of
publications has more then tripled in the years between 2017 and 2020.
Interestingly, this growth in number of publications is not necessarily
reflected in the number of citations. The average number of citations is
consistent for works published in 2017–2020, despite the total number of
articles growing rapidly in these years. A possible explanation lies in the
relatively young age of this research area. Recently published research
articles are consequently very relevant for more recent publications. A clear
illustration of this fact is apparent in the set of publications in 2013,
where only two articles, Ref. [465] and [515], have accumulated a large number
of citations.

Fig. 3 further illustrates how most research is directed towards property
mappings and interatomic potentials. This makes intuitive sense, given that
DFT calculations often underpin large-scale simulations that are based on
data-driven methodologies. Yet, also a growing number of publications is
concerned with applying ML to the electronic structure problem itself.

We perform the same analysis on the two largest categories in Fig. 3
separately. The results are illustrated in Fig. 4. Except for an increasing
interest in applications concerning electrical properties and photovoltaics,
another dominating trend cannot be identified for property mapping
applications. Likewise, it can be seen that while in 2020 nearly half of all
publications using IAPs employed NNs, the relative number of publications for
the respective types of IAPs are relatively evenly distributed throughout the
years. The large number of NN related publications is further illustrated in
Fig. 5, which analyzes the number of publications mentioning certain ML
methods. A prevalence for NNs is apparent, with Ridge Regression and GPR being
the second and third-most used methods, respectively.

Figure 5: Mentions of ML methods throughout the reviewed articles. Note that
only 325 publications could be included in this particular study, as no full
text was available for the remaining 40.

###  IV.2 Conclusions and Outlook

An extensive review over research combining ML and DFT was performed. By
compiling and analyzing over 300 peer-reviewed research articles, we were able
to identify five principal categories of research, presented in Sec. III. Most
current efforts are focused on using data-driven methodologies to assist
conventional DFT studies of materials, either by directly working with
application-specific quantities (Sec. III.1) or by providing ML-IAPs (Sec.
III.2) to facilitate dynamical calculations. The addressed range of topics is
vast and follows pressing scientific problems, such as novel materials
discovery, reducing the human carbon footprint, or enabling novel energy
solutions. A number of publications (see Secs. III.2.5 and III.5) is concerned
with technical aspects of enabling ML-DFT workflows by providing fundamental
software solutions. Of particular importance are databases, which are the
backbone for any data-driven approach. Another, still underrated research area
focuses directly on addressing the shortcomings of DFT as an electronic
structure method. These efforts have been discussed in Sec. III.3. Drawing on
this analysis, a general ML-DFT workflow can be identified, as is shown in
Fig. 2, representing the vast majority of ML-DFT approaches, with only a small
number of very specialized methods not being included. Both the variety as
well as similarities in different methods can be identified in Fig. 2. In
general, the research interest in ML-DFT methods is growing fast, as shown in
Sec. IV.1, and methods addressing the electronic structure problem directly
are gaining traction. Considering the rising number of publication, this
research field is expected to grow further. Future investigations may address
topics such as uncertainty quantification and active learning approaches.
Thereby, they may broaden the utility of ML in computational material science
and computational chemistry.

With ever improving ML-DFT models, new applications that were previously
unattainable will become possible, such as large-scale automated materials
discovery, multi-scale modelling of materials, and digital twins of complex
systems.

## Acknowledgments

This work was funded by the Center for Advanced Systems Understanding (CASUS)
which is financed by the German Federal Ministry of Education and Research
(BMBF) and by the Saxon Ministry for Science, Art, and Tourism (SMWK) with tax
funds on the basis of the budget approved by the Saxon State Parliament. We
thank Aidan Thompson and Sebastian Schwalbe for useful communications.

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