# Proposal for stable atom trapping on a GaN-on-Sapphire chip

Aiping Liu  Institute of quantum information and technology, Nanjing
University of Posts and Telecommunications, Nanjing 210003, China  Broadband
wireless communication and sensor network technology, key lab of Ministry of
Education, Nanjing University of Posts and Telecommunications, Nanjing 210003,
China  Lei Xu  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Xin-Biao Xu  CAS Key Laboratory
of Quantum Information, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Guang-Jie Chen  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Pengfei Zhang  State Key
Laboratory of Quantum Optics and Quantum Optics Devices, and Institute of
Opto-Electronics, Shanxi University, Taiyuan 030006, China  Guo-Yong Xiang
CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Guang-Can Guo  CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
China  CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Qin Wang
<EMAIL_ADDRESS>Institute of quantum information and technology, Nanjing
University of Posts and Telecommunications, Nanjing 210003, China  Broadband
wireless communication and sensor network technology, key lab of Ministry of
Education, Nanjing University of Posts and Telecommunications, Nanjing 210003,
China  Chang-Ling Zou<EMAIL_ADDRESS>CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
China  CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  State Key
Laboratory of Quantum Optics and Quantum Optics Devices, and Institute of
Opto-Electronics, Shanxi University, Taiyuan 030006, China

###### Abstract

The hybrid photon-atom integrated circuits, which include photonic
microcavities and trapped single neutral atom in their evanescent field, are
of great potential for quantum information processing. In this platform, the
atoms provide the single-photon nonlinearity and long-lived memory, which are
complementary to the excellent passive photonics devices in conventional
quantum photonic circuits. In this work, we propose a stable platform for
realizing the hybrid photon-atom circuits based on an unsuspended photonic
chip. By introducing high-order modes in the microring, a feasible evanescent-
field trap potential well $\sim 0.3\,\mathrm{mK}$ could be obtained by only
$10\,\mathrm{mW}$-level power in the cavity, compared with
$100\,\mathrm{mW}$-level power required in the scheme based on fundamental
modes. Based on our scheme, stable single atom trapping with relatively low
laser power is feasible for future studies on high-fidelity quantum gates,
single-photon sources, as well as many-body quantum physics based on a
controllable atom array in a microcavity.

##  I Introduction

With high scalability and enhanced light-matter interactions, the integrated
quantum photonic chip becomes a promising tendency for study of quantum optics
and quantum information processing Kim  _et al._ (2020); Elshaari  _et al._
(2020); Pelucchi  _et al._ (2021). In the past decades, great progress has
been achieved in photonic integrated circuits (PIC), with most attention being
paid to the excellent photonic material platforms Dai  _et al._ (2012);
Bogaerts  _et al._ (2020); Shastri  _et al._ (2021); Siew  _et al._ (2021).
Various quantum photonic devices with passive components and probabilistic
state generation are demonstrated, with the single emitters are strongly
desired for single-photon level optical nonlinearity, which is the essential
resource for quantum light sources and the deterministic quantum operations
Kimble (2008). Neutral atoms with long-lived energy levels provide a potential
solution for the essential single-level nonlinearity to the PIC, which could
also provide the long-coherence time memory for storing quantum information.
Besides, quantum computing and quantum sensing based on atoms can also be
performed with the help of nanophotonic devices Chang  _et al._ (2018);
Garrido Alzar (2019).

Since there is a promising prospect for the photon-atom interaction on a PIC,
considerable efforts have been recently devoted to combining the nanophotonic
devices and neutral atoms to realize efficient quantum devices. Although much
exciting progress has been achieved by the probabilistic strong coupling
between the optical microcavities Dayan  _et al._ (2008); Shomroni  _et al._
(2014); Volz  _et al._ (2014), it is important to realize stable quantum
photonics devices to trap atoms on the nanophotonic devices. Various
approaches have been developed to use near-field optical dipole traps to
confine atoms at the surface of waveguide structures, including the nanofiber
Corzo  _et al._ (2019); Daly  _et al._ (2014); Meng  _et al._ (2020), optical
waveguide Goban  _et al._ (2014); Luan  _et al._ (2020); Béguin  _et al._
(2020); Gehl  _et al._ (2021), the photonic crystal nanocavities Tiecke  _et
al._ (2014); Samutpraphoot  _et al._ (2020); Ðorđević  _et al._ (2021), and
also the microsphere resonator Will  _et al._ (2021). However, these
structures are suspended in a vacuum, so they are potentially sensitive to
vibrations and vulnerable to thermal instability. Alternatively, the atom
trapping by unsuspended waveguide and microring structures have been proposed
Kohnen  _et al._ (2011); Meng  _et al._ (2015); Chang  _et al._ (2019) and
demonstrated Kim  _et al._ (2019); Chang  _et al._ (2020); Zhou  _et al._
(2021). These works of trapping atoms promote the realization of a stable and
scalable hybrid PIC on a substrate for future applications.

In this work, a stable platform for realizing the hybrid photon-atom circuit
based on a suspension-free gallium nitride (GaN) chip is proposed. The
sapphire substrate is transparent to visible lasers, thus allowing efficient
optical access for laser cooling and optical convey belt of cold atoms. To
trap atoms, high-order guided modes in the microring waveguide are used to
obtain a stable atom trapping near the surface of the microring. The required
power of trapping light by hybrid modes trapping is reduced compared to the
fundamental modes trapping. The manipulation of the atom states with waveguide
mode is analyzed numerically, which indicates a strong interaction between the
trapped atom and the confined optical photons through the evanescent field of
waveguide mode. Such a GaN-on-Sapphire platform is feasible for hybrid photon-
atom circuits and is promising for various quantum photonic devices.

Figure 1: Illustration of the microring platform for photon-atom interaction.
(a) The microring platform with the atoms loaded to the surface of microring
by the free space laser beam. (b) The microring platform with the atoms
trapped on the surface of microring by the optical evanescent fields formed by
the guiding modes.

##  II The platform for hybrid photon-atom circuit

The proposed experimental platform for hybrid photon-atom circuits is
illustrated in Fig. 1. The platform is made of GaN microrings and waveguides
on a sapphire substrate, which is transparent to visible light. Such a
platform has been extensively studied over the past decades, showing its
excellent optical properties in visible wavelengths Zhang  _et al._ (2011);
Bruch  _et al._ (2015); Chen  _et al._ (2017); Li  _et al._ (2018); Stassen
_et al._ (2019a) and also its potential in photonic integrated circuits Fu
_et al._ (2019); Xu  _et al._ (2022). Distinct from the conventional platforms
for photon-atom interaction on a silicon substrate, all the photonic
microstructures in our system are fabricated on the substrate, without any
suspended structure or any fabrication procession of the substrate. The hybrid
photon-atom circuits are prepared in three steps:

(i) Place the GaN-on-sapphire chip in vacuum. Since all the materials are
transparent to visible light, the conventional magneto-optical trap could be
implemented by transmitting laser beams through the chip. A cloud of cold
atoms could be prepared by Doppler cooling, with the atomic cloud being only a
few $mm$ away from the top suface of the microring device.

(ii) Guide the cold atoms to the chip. By sending a pair of counter-
propagating focused Gaussian beams of red-detuned laser vertically through the
transparent substrate, with the waist of the laser beams close to the
microring on the chip, the cold atoms could be trapped and transported to the
top surface of the microring by the optical conveyor belt Schrader  _et al._
(2001); Kuhr  _et al._ (2001). As illustrated in Fig. 1(a), by tuning the
phase of upward and downward dipole laser beams, atoms could be guided and
transported to the chip.

(iii) When the cold atoms are guided to a certain distance from the top
surface of the microring, turn on the blue- and red-detuned lasers in the
photonic integrated circuits, thus creating the two-color optical dipole trap
Le Kien  _et al._ (2004); Vetsch  _et al._ (2010); Fu  _et al._ (2007); Chang
_et al._ (2019) via the evanescent fields of the microring resonator. Then,
the single atoms could be confined to the microring resonator, as shown in
Fig. 1(b), and coupled to the signal photons near the transition frequency of
the atoms.

Following this procedure, our platform offers a basic cavity-QED system for
realizing the atom-photon quantum gates, atom-mediated photon-photon
entanglement, as well as quantum storage. It is also worth noting that the
materials of this platform hold several advantages: Firstly, the unsuspended
photonic circuit platform is very stable and is robust to the external
mechanical perturbations and the thermal instability. Especially, to provide a
deep trap for atoms by the evanescent field, a strong laser intensity in the
microring should be excited, which will result in serious heating Padmaraju
_et al._ (2012); Padmaraju and Bergman (2014). Secondly, since the GaN and
sapphire are wide-bandgap (0.78-4.77 eV) materials transparent to a ultra-
broad band wavelengths (260 nm-1590 nm) Yu  _et al._ (1997); Muth  _et al._
(1999), thus the scheme is valid for most atoms working with visible and near-
visible lasers. Thirdly, for atomic transition at visible wavelengths, the
refractive indices of the GaN and substrate are around $n_{1}=2.2$ and
$n_{2}=1.7$, respectively. The large refractive index contrast indicates
strong optical confinement in the GaN microphotonic structures Zheng  _et al._
(2020). Lastly, GaN is not only promising for nonlinear photonics with large
intrinsic nonlinearities and a large bandgap, but also quite resilient to high
temperature and optical power. So the GaN microring can sustain a relative
high intracavity optical power for the efficient manipulation of the neutral
atoms.

As all the essential components and technique of the hybrid photon-atom
circuit are feasible for experiments, here we are focusing on the design of
the photonic structure for reasonable dipole traps and also for strong photon-
atom interactions. So, in the following, we will investigate the performance
of the dipole trap in detail and the potential photon-atom coupling strength.

##  III The simulation of the trap potential well

In this work, we consider the Rubidium atoms with the D2 transition wavelength
of 87Rb atoms $\lambda_{0}=780\,\mathrm{nm}$. The optical properties of the
GaN microring are determined by the geometry, which could be described by the
major radius $R$, width $w$ and height $h$ of the cross-section of the
microring, as shown in the up part of Fig. 2(a) , where $r=10\,\mathrm{\mu
m},w=600\,\mathrm{nm}$, and $h=280\,\mathrm{nm}$. Two lasers, with the
wavelengths $\lambda_{b}=760\,\mathrm{nm}$ and $\lambda_{r}=852\,\mathrm{nm}$,
respectively, are serving as the blue- and red-detuned dipole trap light Le
Kien  _et al._ (2004); Vetsch  _et al._ (2010); Fu  _et al._ (2007); Chang
_et al._ (2019). When the two lasers are loaded into the microring, the
excited microring modes could enhance the electric field and their evanescent
parts in vacuum to offer the trap potential. The evanescent field decays
exponentially away from the surface of the microring, so it provides a
repulsive and a attractive force on the considered 87Rb atom by the blue- and
red-detuned light modes, respectively. In addition, the attractive van der
Waals force increases fast when the atom is close to the surface of the
microring, thus the atoms will stick to the surface. Benefiting from the
resonance enhancement of the microring, a strong evanescent field can be
obtained by a relatively weak laser input to compensate for the van der Waals
force. By appropriately manipulating the incident light, the dipole trap
potential of the evanescent field and the van der Waals potential will form a
stable potential well for trapping 87Rb atoms. For a balance between these
optical forces, a trap position $100\,\mathrm{nm}$ away from the microring
surface is considered in the following studies.

Figure 2: The optical trap formed with the fundamental mode of microring.(a)
The cross-section of the microring (up) and the optical field distributions of
the propagating modes in the microring (down). (b) The optical potential
distribution formed by the fundamental modes in the $r-z$ plane above the
microring surface. (c) The optical traps vary with
$P_{\mathrm{r}}/P_{\mathrm{b}}$ for different $P_{\mathrm{b}}$.

There are different waveguide modes in the microring waveguide, as illustrated
in the inset of Fig. 2(a) is the electric field distributions of the
fundamental modes $\mathrm{TM}_{0}$ and $\mathrm{TE}_{0}$, as well as high-
order modes $\mathrm{TM}_{1}$ and $\mathrm{TE}_{1}$ modes Ji  _et al._ (2021)
for incident light wavelength $\lambda_{b}$. Compared with $\mathrm{TE}$
modes, $\mathrm{TM}$ modes have a stronger evanescent field on the top surface
of waveguide, so only $\mathrm{TM}$ modes are considered in the following.
Shown in Fig. 2(b) is the cross section of simulated trap potential formed on
top of the microring, in the simplest case that fundamental mode
$\mathrm{TM}_{0}$ is excited for blue- and red-detuned lasers. Similar to the
pioneer studies by Ref. Chang  _et al._ (2019), the trap potential well is a
narrow crescent because of the asymmetric field distributions of the microring
modes on the microring cross section. The center of the trap potential well is
marked by the red cross “$+$" and the saddle point is marked by the black
cross “$\times$", respectively. An effective trap potential of
$0.265\,\mathrm{mK}$ (the potential difference between the well and saddle
points) is obtained on the designed platform with $150\,\mathrm{mW}$ blue-
detuned and $195\,\mathrm{mW}$ red-detuned $\mathrm{TM}_{0}$ laser power
inside the microring. It is worth noting that the potential is evaluated for a
given intracavity power, which should be much higher than the input power in
the waveguide due to the resonant enhancement by the microring cavity modes.
For example, with a practical quality factor of $Q\approx 2\times 10^{6}$ and
a free-specral range of $FSR\approx 320\,\mathrm{GHz}$ Zheng  _et al._ (2020);
Stassen  _et al._ (2019b), the required input laser power to the microring
could be only about $0.4\,\mathrm{mW}$.

The dependence of optical trap depth $U_{\mathrm{eff}}$ on the power of the
circulating laser power inside the microring is shown in Fig. 2(c), where
$P_{\mathrm{b}}$ and $P_{\mathrm{r}}$ are the powers of the blue- and red-
detuned $\mathrm{TM}_{0}$ modes circulating in the waveguide, respectively.
The curves illustrate the optical trap depth $U_{\mathrm{eff}}$ varying with
the power ratio of red- and blue-detuned $\mathrm{TM}_{0}$ modes
($P_{\mathrm{r}}/P_{\mathrm{b}}$), and the different curves correspond to the
power of the blue-detuned $\mathrm{TM}_{0}$ mode as $100\,\mathrm{mW}$,
$120\,\mathrm{mW}$, $140\,\mathrm{mW}$, $160\,\mathrm{mW}$,
$180\,\mathrm{mW}$, and $200\,\mathrm{mW}$, respectively. From Fig. 2(c), the
optical trap depth is not only increased with the $P_{\mathrm{b}}$, but also
related to the $P_{\mathrm{r}}/P_{\mathrm{b}}$. With a constant
$P_{\mathrm{b}}$, the optical trap depth $U_{\mathrm{eff}}$ increases at first
and then decreases when the $P_{\mathrm{r}}/P_{\mathrm{b}}$ increases. In
fact, the optical trap can not be formed when the
$P_{\mathrm{r}}/P_{\mathrm{b}}$ is too small or too large especially with low
mode power. Besides of high enough intracaity powers, an appropriate
$P_{\mathrm{r}}/P_{\mathrm{b}}$ is also required to form a large effective
optical trap depth $U_{\mathrm{eff}}$.

Figure 3: The optical trap formed with hybrid modes. (a) The optical trap
potential well above the microring surface in the $r-z$ plane. (b) The optical
trap potential well formed with hybrid modes combined with bidirectional red-
detuned $\mathrm{TM}_{0}$ modes and unidirectional blue-detuned
$\mathrm{TM}_{0}$ and $\mathrm{TM}_{1}$ modes in the $r-\phi$ plane with
$100\,\mathrm{nm}$ above the microring surface. Figure 4:  (a) The optical
traps vary with $P_{\mathrm{r}}/P_{\mathrm{b}}$ for different $P_{\mathrm{b}}$
with $P_{\mathrm{b1}}=1.6\,\mathrm{mW}$.(b) The optical traps vary with
$P_{\mathrm{r}}/P_{\mathrm{b}}$ for different $P_{\mathrm{b1}}$ with
$P_{\mathrm{b}}=15\,\mathrm{mW}$. The inset gives the relation between the
optical trap and $P_{\mathrm{b1}}$ with $P_{\mathrm{b}}=15\,\mathrm{mW}$ and
$P_{\mathrm{r}}/P_{\mathrm{b}}=1.335$.

From the trap potential distribution in Fig. 2(b), the effective trap depth is
limited by the relative low trap depth at the saddle point. In order to
further reduce the required laser power, we should improve the trap depth at
the saddle point and obtain a more regular optical trap potential well.
Therefore, a set of modes composed of blue-detuned $\mathrm{TM}_{0}$ mode,
red-detuned $\mathrm{TM}_{0}$ mode and blue-detuned $\mathrm{TM}_{1}$ mode are
used to form the optical trap potential well. As shown in Fig. 3(a) is the
optical trap potential well formed with these hybrid modes, where the saddle
point has a much higher potential barrier because the bivalve field of the
assisted $\mathrm{TM}_{1}$ mode makes up the weak field on the two sides of
the $\mathrm{TM}_{0}$ mode. The optical trap depth, with the trap center
marked by the red cross “$+$" and the saddle point marked by the black cross
“$\times$", is $0.272\,\mathrm{mK}$ with $P_{\mathrm{b}}=15\,\mathrm{mW}$,
$P_{\mathrm{r}}/P_{\mathrm{b}}=1.335$, and the blue-detuned $\mathrm{TM}_{1}$
mode power $P_{\mathrm{b1}}=1.6\,\mathrm{mW}$. Further, the relations between
the $U_{\mathrm{eff}}$ and the $P_{\mathrm{r}}/P_{\mathrm{b}}$ for different
$P_{\mathrm{b}}$ are provided in Fig. 4(a). The total mode power in the hybrid
modes trapping scheme is reduced by one order of magnitude compared with the
trapping scheme using the fundamental modes. With reduced incident light
power, the potential thermal effect or mechanical vibration of the microring
will be greatly suppressed, which makes the optical trapping more stable. Then
a stable atom trap can be realized on the plane perpendicular to the microring
waveguide.

The relation between the optical trap depth $U_{\mathrm{eff}}$ and the
$P_{\mathrm{r}}/P_{\mathrm{b}}$ with $P_{b}=15\,\mathrm{mW}$ is given in Fig.
4(b), where different curves correspond to different $P_{\mathrm{b1}}$. With
increased $P_{\mathrm{r}}/P_{\mathrm{b}}$, the optical trap depth
$U_{\mathrm{eff}}$ increases at first and then decreases after a maximun
$U_{\mathrm{eff}}$. From different curves, the optical trap depth
$U_{\mathrm{eff}}$ is larger with a bigger $P_{\mathrm{b1}}$, so is the
maximun $U_{\mathrm{eff}}$. The inset of Fig. 4(b) gives the relation between
the optical trap depth and the blue-detuned $\mathrm{TM}_{1}$ mode power
$P_{\mathrm{b1}}$ with $P_{\mathrm{b}}=15\,\mathrm{mW}$ and
$P_{\mathrm{r}}/P_{\mathrm{b}}=1.335$. As the power of the blue-detuned
$\mathrm{TM}_{1}$ mode $P_{\mathrm{b1}}$ increases, the optical trap depth
increases fast at first to reach a maximum, and then decreases slowly. Thus,
an appropriate $P_{\mathrm{b}}$ makes the optical trap depth more efficient
and regular, thus stable and require low power consumption.

In addition, the bidirectional red-detuned $\mathrm{TM}_{0}$ modes are used to
form the stationary field with other microring modes as unidirectional
circulating waves. The optical trap potential well formed on top of the
microring is given in Fig. 3(b). It should be noted that, because of the
material dispersion the wavevectors of the blue- and red-detuned circulating
modes are different, only red-detuned $\mathrm{TM}_{0}$ mode is bidirectional
circulating. Thus a stable atom trapping in three dimensions can be realized
by the hybrid modes trapping on the platform based on suspension-free GaN-on-
sapphire chip.

##  IV Photon-atom interaction

Figure 5: The performance of microcavity enhanced atom-photon interaction of
${}^{8}7$Rb in the trap center T($r=R$, $z-h=100\,\mathrm{nm}$). (a) The mode
area $A_{m}$, and coupling strength $g$ (b) cooperativity $C$ vary with the
waveguide width $w$ of the microring with radius $R=10\,\mathrm{\mu m}$ and
$h=280\,\mathrm{nm}$. (c) The mode area $A_{m}$, and coupling strength $g$ (d)
cooperativity $C$ vary with the waveguide height $w$ of the microring with
$R=10\,\mathrm{\mu m}$ and $w=600\,\mathrm{nm}$.

The evanescent field around the microring waveguide will interact with the
atoms trapped around the microring waveguide. Benefited from the low loss of
GaN and excellent processing technology, the photonic loss rate ($\kappa$) can
be smaller than the photon-atom coupling strength $g$, so is the atomic
radiative decay rate ($\gamma$) into free space. Thus, the photon-atom strong
coupling regime is potentially achievable in current platform. For a perfect
quantum emitter, the photon-atom coupling strength can be given as Corwin  _et
al._ (1999); Chang  _et al._ (2019)

$g=\sqrt{\frac{3\lambda^{3}\omega_{0}\gamma}{16\pi^{2}V_{m}}},$ (1)

where $\lambda$ is the freespace wavelength, $\omega_{0}$ is the frequency of
the photon mode. $\gamma/2\pi\approx 6.1\,\mathrm{MHz}$ is the spontaneous
emission rate of the atom in vacuum, i.e. equal to the angular frequency of
natural linewidth. $V_{m}$ denotes the effective mode volume and could be
derived approximately $V_{m}\approx 2\pi RA_{m}$ for the microring structure
with a radius of $R$, and the $A_{m}$ is the effective mode area. For an atom
located at working point $(r_{0},z_{0})$,

$A_{m}(r,z)=\frac{\int\varepsilon(r,z)\left|E(r,z)\right|^{2}drdz}{\varepsilon(r_{0},z_{0})\left|E(r_{0},z_{0})\right|^{2}},$
(2)

where $E(r,z)$ is the electric field distribution and $\varepsilon$ is the
distribution of dielectric permittivity.

Therefore, the merit for quantum coherence of the microcavity-enhanced atom-
photon interaction can be figured by the cooperativity parameter as

$C=\frac{g^{2}}{\kappa\gamma}=\frac{3\lambda^{3}}{4\pi^{2}}\frac{Q}{V_{m}},$
(3)

where $\lambda$ is the wavelength of light in free space. The cooperativity
parameter mainly depends on the ratio of quality factor $Q$ and effective mode
volume $V_{m}$. High cooperativity is pursued for the high-fidelity quantum
gate between atoms and photons, and also potential nonlinear optical effects
at single-photon levels. Also, the enhanced photon-atom interaction could also
be applied to the single-photon sources, with the cavity enhanced collection
efficiency of the radiation from the atomic quantum emitter

$\eta=\frac{C}{1+C}.$ (4)

Note that $C$ is equivalent to the Purcell factor, and the estimation of the
$\eta$ is only valid in the bad-cavity limit with $g<\kappa$ Paul R. Berman
(1994). If $g$ is too strong, we could increase the cavity linewidth by
introducing a higher external coupling rate to the microring cavity to meet
the bad-cavity limit.

As studied in the above section, the two-color optical dipole trap could be
constructed, and it is feasible to trap single atoms on top of the waveguide
around $T(r=R,z=100\,\mathrm{nm})$. We are targeting at the strong atom-photon
interaction, thus we are focusing on the fundamental mode polarized along $z$
direction at the wavelength of D2 transitions of Rubidium atom ($\sim
780\,\mathrm{nm}$). For the GaN microring with a radius of a few tens of
micrometer, the quality factor $Q$ is mainly decided by the material
absorption and scattering loss, and $Q=2\times 10^{6}$ with the corresponding
$\kappa/2\pi\approx 100\,\mathrm{MHz}$ is taken in the simulations Zheng  _et
al._ (2020); Stassen  _et al._ (2019b).

Figure 5 summarizes the performance of the microcavity enhanced atom-photon
interaction in the trap center T($r=R$, $z-h=100\,\mathrm{nm}$) above the
microring with different geometry parameters $w$ and $h$. The effective mode
area $A_{m}$ and the photon-atom coupling strength $g$ for the 87Rb and the
microring with $R=10\,\mathrm{\mu m}$ and $h=280\,\mathrm{nm}$ are given in
Fig. 5(a). The effective mode area $A_{m}$ increases with increased waveguide
width, since a wide waveguide constraints more mode field inside the waveguide
with less mode field outside. With an increased effective mode area $A_{m}$,
the photon-atom coupling strength $g$ decreases. The cooperativity parameter
$C$ of the 87Rb atom also decrease with increased waveguide width $w$ of
microring as shown in Fig. 5 (b). The photon-atom interaction is also related
to the waveguide height as shown in Fig. 5(c) and (d) for the microring with
$R=10\,\mathrm{\mu m}$ and $w=600\,\mathrm{nm}$. The effective mode area
$A_{m}$ increases and the photon-atom coupling strength $g$ decreases as the
waveguide height $h$ increases, since thicker waveguide has less evanescent
field outside. The cooperativity parameter C of the 87Rb atom also decreases
as the waveguide height $h$ increases.

Figure 6: The cut-off condition of the $\mathrm{TM_{1}}$ mode for the blue-
and red-detuned wavelength. The lines indicate the crictical parameters for
the cut-off condition, and the mode is allowed only when $h$ and $w$
parameters are above the critical line.

From the above analysis, the photon-atom interaction increases as the size of
microring waveguide decreases. In the same time, the mode loss of the
microring cavity will increase as the size of the waveguide decreases. So, an
appropriate size of the microring waveguide is important to guarantee the
efficient photon-atom interaction. Table 1 gives the photon-atom interaction
in the optical trap center T($r=R$, $z-h=100\,\mathrm{nm}$) above the GaN
microring waveguide with different parameters incident with
$P_{\mathrm{b}}=15\,\mathrm{mW}$ and $P_{\mathrm{b1}}=1.8\,\mathrm{mW}$. It is
shown that the depth of optical trap well is larger than 0.13 $\mathrm{mk}$,
which can trap single atom stably. In the same time, the trapped atom has an
efficient photon-atom interaction with microring waveguide modes. The photon-
atom coupling strength $g/2\pi$ is larger than $110\,\mathrm{MHz}$ and the
cooperativity parameter $C$ is larger than 42 with the corresponding
collection efficiency above $97.7\,\%$. Since the $\mathrm{TM_{b,1}}$ mode is
used to improve the trap of the atom, the cut-off of the $\mathrm{TM_{1}}$
mode is provided in Fig. 6. For small waveguide width $w$ and height $h$, the
${\mathrm{TM_{1}}}$ mode can be excited and propagate well in the microring
waveguide. So the proposed platform based on suspension-free GaN-on-sapphire
chip is feasible to provide stable atom trapping with hybrid modes for the
photon-atom interaction in the integrated quantum information.

Table 1: The photon-atom interaction of 87Rb on the trap well formed above the GaN microring waveguide incident with intracavity trap laser power $P_{\mathrm{b}}=15\,\mathrm{mW}$, $P_{\mathrm{b1}}=1.8\,\mathrm{mW}$, microring radius $R=10\,\mathrm{\mu m}$. $w\,(\mathrm{\mu m})$ | 0.7 | 0.6 | 0.6 | 0.6 | 0.6
---|---|---|---|---|---
$h\,(\mathrm{\mu m})$ | 0.28 | 0.32 | 0.3 | 0.28 | 0.28
$P_{\mathrm{r}}/P_{\mathrm{b}}$ | 1.27 | 1.25 | 1.25 | 1.32 | 1.36
$\begin{array}[]{c}T(r-R,z-h)\\\ (\mathrm{nm},\mathrm{nm})\end{array}$ | (1,107) | (-4,103) | (-28,97) | (8,102) | (3,102)
$U_{\mathrm{min}}\,(\mathrm{mK})$ | 0.265 | 0.205 | 0.237 | 0.336 | 0.395
$U_{\mathrm{eff}}\,(\mathrm{mK})$ | 0.197 | 0.131 | 0.141 | 0.249 | 0.264
$A_{\mathrm{m}}\,(\mathrm{\mu m}^{2})$ | 23.5 | 27.4 | 20.0 | 17.1 | 14.7
$g/2\pi$ (MHz)) | 118.7 | 110.0 | 128.6 | 139.2 | 150.1
$C$ | 48.9 | 42 | 57.4 | 67.2 | 78.2
$\eta\,(\%)$ | 98.0 | 97.7 | 98.3 | 98.5 | 98.7

##  V Conclusion

A stable platform for trapping atoms is proposed to realize the hybrid photon-
atom circuit based on a suspension-free GaN-on-sapphire chip. The high-order
waveguide modes are used for realizing a stable trap potential well with low
incident trap light power, with which a feasible and regular dipole trap
potential well around $300\,\mathrm{\mu K}$ can be obtained by only
$10\,\mathrm{mW}$-level power circulating in the microring (equivalently sub-
mW-level input power in the coupling waveguide). In addition, the interaction
between trapped atom and guided photon in the resonator is analyzed to confirm
a high-cooperative photon-atom interface for potential applications. Such a
stable on-chip atom trapping scheme will play a key role in the hybrid photon-
atom circuit, which provides a unique experimental platform for studying the
quantum optics effects with novel photonic structure, the single-photon level
nonlinear photonics, and the quantum devices for scalable quantum information
processing Chang  _et al._ (2018).

##  VI Acknowledgments

This work was supported by the National Key Research and Development Program
of China (Grant Nos. 2018YFA0306400, 2017YFA0304100), the National Natural
Science Foundation of China (Grant Nos. 11922411, U21A6006, 12104441,
12134014, 12074194), the Project funded by China Postdoctoral Science
Foundation (SBH190004), the Leading-edge technology Program of Jiangsu Natural
Science Foundation (Grant No. BK20192001), and the Fund for Shanxi “1331
Project” Key Subjects Construction. C.-L.Z. was also supported by the
Fundamental Research Funds for the Central Universities, and the Program of
State Key Laboratory of Quantum Optics and Quantum Optics Devices. This work
was partially carried out at the USTC Center for Micro and Nanoscale Research
and Fabrication.

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# A deep neural network approach on solving the linear transport model under
diffusive scaling

Liu Liu, Tieyong Zeng , Zecheng Zhang Department of Mathematics, The Chinese
University of Hong KongDepartment of Mathematics, The Chinese University of
Hong KongDepartment of Mathematics, Purdue University, West Lafayette, IN
47906, USA

###### Abstract

In this work, we propose a learning method for solving the linear transport
equation under the diffusive scaling. Due to the multiscale nature of our
model equation, the model is challenging to solve by using conventional
methods. We employ the physical informed neural network (PINN) framework, a
mesh-free learning method that can numerically solve partial differential
equations. Compared to conventional methods (such as finite difference or
finite element type), our proposed learning method is able to obtain the
solution at any given points in the chosen domain accurately and efficiently,
which enables us to better understand the physics underlying the model. In our
framework, the solution is approximated by a neural network that satisfies
both the governing equation and other constraints. The network is then trained
with a combination of different loss terms. Using the approximation theory and
energy estimates for kinetic models, we prove theoretically that the total
loss vanishes as the neural network converges, upon which the neural network
approximated solution converges pointwisely to the analytic solution of the
linear transport model. Numerical experiments for two benchmark examples are
conducted to verify the effectiveness and accuracy of our proposed method.

##  1 Introduction

The linear transport model, which describes the kinetic particles collision
and absorption through a material medium while evolving in time, is an
important physics model and arises in applications of various fields. In this
paper, we study the linear transport equation under the diffusive scaling by
using the deep neural network approach.

The deep learning method has been successful in solving various problems in
the computer science area; it improves the computation speed and the accuracy
of many tasks such as the image classification, segmentation and language
processing. Once the model is trained, one can use it by substituting the
testing samples in the model, which usually is a forward process and only
tensor productions are involved. The data-driven solvers for partial
differential equations (PDEs) have drawn an increasing attention due to their
capability to encode the underlying physical laws governed by the model
equation and give relatively accurate predictions for the unknowns. In
particular, we mention the physical informed neural networks (PINNs) framework
[10, 18, 16, 11, 19, 7, 21]. This method leverages the benefits of auto-
differentiation of the current software and the underlying physics of the
PDEs, which can be incorporated into the network by minimizing the losses
composed of the PDE’s residuals and other constraints such as the initial and
boundary conditions. There are many other works developed, for example [2, 3,
26, 24]. In this work, we apply this popular method to solve the important,
hyperbolic-type kinetic equation.

The Deep Learning method can resolve some difficulties in traditional finite
difference-type numerical methods [17], such as the expensive computational
cost especially for problems with high-dimensional physical variables, the
challenge dealing with complex boundary conditions, truncation of the velocity
domain (in some models such as the Boltzmann equation, the velocity lies in
the three-dimensional whole space thus a truncation is needed in numerical
discretizations). Besides, the Deep Learning algorithm has the advantage of
being intuitive and easy to be executed. For example, instead of designing
mass (or momentum and energy, if applicable) conservative schemes for kinetic
models, that are challenging in traditional numerical methods [6], one can
simply involve the derivative in time of conserved quantities of interests in
the total loss function, as implemented in [8].

We mention some other advantages of using the DNN approach to solve the linear
transport model: a) to obtain the distribution function at any given $t$, $x$,
$v$, instead of getting only discrete values at the uniform mesh in
traditional finite volume or finite element numerical methods; as a mesh-free
method, works efficiently for high-dimensional physical space problems; b) to
avoid high computational cost on the simulation of kinetic equations due to
the velocity variable $v$, and the integral-based, nonlocal collision
operators appeared in other complicated kinetic models such as the Boltzmann
or Landau equations. Nevertheless, we mention that there are indeed some
weaknesses of the Deep Learning approach. First, there is no guarantee that
the Deep Learning algorithm will converge and it is practically difficult to
show their convergence. It is also hard to evaluate the accuracy of the DNN
approach in contrast with traditional numerical methods.

This paper is organized as the following. In Section 2, we introduce the
background of the linear transport model under the diffusive scaling. In
Section 3, we review and discuss the neural network framework and method for
solving general PDEs. Two main convergence results are given in Section 4,
showing that 1) the loss function goes to zero as the neural network
converges; 2) the neural network solution converges point-wisely to the
analytic solution when the loss function converges to zero. The effectiveness
and accuracy of our proposed method, including the choice of weights in the
total loss function will be presented in Section 5. Finally, we summarize the
paper and mention some future work in Section 6.

##  2 The linear transport model

The linear transport model, which describes the kinetic particles collision
and absorption through a material medium while evolving in time, arises in
many applications, such as atmosphere and ocean modeling [4, 20, 25],
astrophysics [15] or nuclear physics. Such problems usually involve several
orders of magnitudes of length scales characterized by the Knudsen number,
defined as the ratio of the mean free path over a typical length scale such as
the size of the spatial domain.

We consider the linear transport equation under diffusive scaling, with one-
dimensional space and velocity variable ($x\in\Omega\subset\mathbb{R}$,
$v\in[-1,1]$) and given by

$\displaystyle\begin{split}&\varepsilon\partial_{t}f+v\cdot\nabla_{x}f=\frac{1}{\varepsilon}\mathcal{L}(f),\qquad
f(t=0,x,v)=f_{0}(x,v),\\\\[4.0pt]
&\mathcal{L}(f)=\sigma(x)\left[\frac{1}{2}\int_{-1}^{1}f(t,x,v)dv-f(t,x,v)\right].\end{split}$
(2.1)

##  3 The Neural Network approach

We briefly review the Deep Neural Network (DNN) structure and approach
introduced in [8], where the one-dimensional kinetic Fokker-Planck equation is
studied. Denote the approximated function as $f^{nn}(t,x,v,z;m,w,b)$ and
suppose DNN has $L$ layers; the input layer takes $(t,x,v)$ as input and the
final layer gives $f^{nn}(t,x,v,z;m,w,b)$ as the output. The relation between
the $l$-th and $(l+1)$-th layer ($l=1,2,\cdots L-1)$ is given by

$u_{j}^{(l+1)}=\sum_{i=1}^{m_{l}}w_{ji}^{(l+1)}\bar{\sigma}_{l}(z_{i}^{l})+b_{j}^{(l+1)},$

where $m=\left(m_{0},m_{1},m_{2},\dots,m_{L-1}\right)$, the weights
$w=\left\\{w_{ji}^{(k)}\right\\}_{i,j,k=1}^{m_{k-1},m_{k},L}$ and the bias
$b=\left\\{b_{j}^{(k)}\right\\}_{j=1,k=1}^{m_{k},L}$ are given in [8], which
we refer to for details.

Regarding the optimization algorithm, we use Adam optimization algorithm,
which is an extended algorithm of the stochastic gradient descent and is
popularly used in the applications of the deep learning.

###  3.1 Definition of loss functions

The loss function for the governing linear transport equation (2.1) is defined
by

$\displaystyle\begin{split}&\displaystyle\quad Loss_{GE}\\\\[4.0pt]
&\displaystyle=\int_{0}^{T}\int_{\Omega}\int_{V}\Big{|}\varepsilon\partial_{t}f^{nn}(t,x,v;m,w,b)+v_{k}\partial_{x}f^{nn}(t,x,v;m,w,b)\\\\[4.0pt]
&\displaystyle\quad-\frac{1}{\varepsilon}\mathcal{L}(f^{nn})(t,x,v;m,w,b)\Big{|}^{2}\,dvdxdt\\\\[4.0pt]
&\displaystyle\approx\frac{1}{N_{i,j,k}}\sum_{i,j,k}\Big{|}\partial_{t}f^{nn}(t_{i},x_{j},v_{k};m,w,b)+v_{k}\partial_{x}f^{nn}(t_{i},x_{j},v_{k};m,w,b)\\\\[4.0pt]
&\displaystyle\quad-\frac{1}{\varepsilon}\mathcal{L}(f^{nn})(t_{i},x_{j},v_{k};m,w,b)\Big{|}^{2},\end{split}$
(3.1)

where $N_{i,j,k}=N_{i}N_{j}N_{k}$, and the collision operator

$\mathcal{L}(f^{nn})(t_{i},x_{j},v_{k};m,w,b)=\sigma(x_{j})\left[\frac{1}{2N_{k}}\sum_{k}f^{nn}(t_{i},x_{j},v_{k};m,w,b)-f^{nn}(t_{i},x_{j},v_{k};m,w,b)\right].$

Define $n_{x}$ the unit outward normal vector on the boundary
$\partial\Omega$, and $\gamma\stackrel{{\scriptstyle\text{ def
}}}{{=}}\partial\Omega\times[-1,1]$. This phase boundary can be split into an
outgoing boundary $\gamma_{+}$, incoming boundary $\gamma_{-}$ and a singular
boundary $\gamma_{0}$, defined by

$\displaystyle\gamma_{+}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\left\\{(x,v)\in\partial\Omega\times[-1,1]:n_{x}\cdot v>0\right\\},$
$\displaystyle\gamma_{-}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\left\\{(x,v)\in\partial\Omega\times[-1,1]:n_{x}\cdot v<0\right\\},$
$\displaystyle\gamma_{0}\stackrel{{\scriptstyle\text{ def
}}}{{=}}\left\\{(x,v)\in\partial\Omega\times[-1,1]:n_{x}\cdot v=0\right\\}.$

We now define the loss terms for the initial condition and the boundary
conditions:

$\displaystyle\begin{split}\displaystyle
Loss_{IC}&=\int_{\Omega}\int_{V}\left|f^{nn}(0,x,v)-f_{0}(x,v)\right|^{2}\,dvdx\\\\[4.0pt]
\displaystyle&\approx\frac{1}{N_{j,k}}\sum_{j,k}\left|f^{nn}(0,x_{j},v_{k})-f_{0}(x_{j},v_{k})\right|^{2}.\end{split}$
(3.2)

For inflow boundary condition $f(t,x,v)|_{\gamma^{-}}=g(t,x,v)$ on
$x\in\partial\Omega$, one has

$\displaystyle\begin{split}\displaystyle
Loss_{BC}&=\sum_{x\in\partial\Omega}\int_{0}^{T}\int_{V}\left|f^{nn}(t,x,v;m,w,b)-g(t,x,v;m,w,b)\right|^{2}\,dvdt\\\\[4.0pt]
\displaystyle&\approx\frac{1}{|\partial\Omega|N_{i,k}}\sum_{i,k}\left|f^{nn}(t_{i},x,v_{k};m,w,b)-g(t_{i},x,v_{k};m,w,b)\right|^{2},\end{split}$
(3.3)

where $|\partial\Omega|$ denotes the volume of the spatial boundary. Add up
(3.1), (3.2) and (3.3), with appropriate weights
$\\{\lambda_{g},\lambda_{i},\lambda_{b}\\}$, the total loss function is
defined by

$Loss_{Total}=\lambda_{g}\,Loss_{GE}+\lambda_{i}\,Loss_{IC}+\lambda_{b}\,Loss_{BC}.$
(3.4)

##  4 Analysis results

In this section, we show the two main theoretical results.

Recall that neural network architecture was first introduced in [14]. Later in
[5], Cybenko established sufficient conditions where a continuous function can
be approximated by finite linear combinations of single hidden layer neural
networks, followed by the work in [12] that extends the theory to the multi-
layer network case. We paraphrase the Universal Approximation Theorem [5] to
the form that is needed in our context: If $f$ solves the linear transport
model (2.1) and is sufficiently smooth in all variables, then there exists a
two-layer neural network

$f^{nn}(t,x,v)=\sum_{i=1}^{m_{1}}w_{1i}^{(2)}\bar{\sigma}\left(\left(w_{i1}^{(1)},w_{i2}^{(1)},w_{i3}^{(1)}\right)\cdot(t,x,v)+b_{i}^{(1)}\right)+b_{1}^{(2)}$

such that

$||f-f^{nn}||_{L^{\infty}(K)}<\eta,\quad||\partial_{t}(f-f^{nn})||_{L^{\infty}(K)}<\eta,\quad||\nabla_{x}(f-f^{nn})||_{L^{\infty}(K)}<\eta,$

where the domain $K=[0,T]\times\Omega\times[-1,1]$. Note that our analysis
results in this section apply to the neural network with multiple layers, due
to [12]. For simplicity, we review the Universal Approximation Theorem in its
two-layer form.

###  4.1 Convergence of the loss function

We first show that a sequence of neural network solutions to (2.1) exists such
that the total loss function converges to zero.

######  Theorem 4.1.

Let $f$ solves the equation (2.1) and satisfies $f\in C^{1}([0,T])\cap
C^{1}(\Omega\times V)$. Then there exists a sequence of neural network
parameters $\\{m_{[j]},w_{[j]},b_{[j]}\\}_{j=1}^{\infty}$ such that the
sequence of DNN solutions with $m_{[j]}$ nodes, given by
$\\{f_{j}(t,x,v)=f^{nn}(t,x,v;m_{[j]},w_{[j]},b_{[j]})\\}_{j=1}^{\infty}$
satisfies

$Loss_{\text{Total}}(f_{j})\to 0,\qquad\text{as }j\to\infty.$

Proof. Denote the velocity domain $\mathcal{D}=[-1,1]$ and
$\mu(V)=\int_{\mathcal{D}}dv=2$. Define

$d_{ge,j}(t,x,v,z):=-\left[\varepsilon\partial_{t}+v\cdot\nabla_{x}\right]f_{j}+\frac{1}{\varepsilon}\mathcal{L}(f_{j}).$
(4.1)

Integrate $|d_{ge,j}|^{2}$ over $[0,T]\times\Omega\times\mathcal{D}$, which is
equivalent to

$|\varepsilon\partial_{t}(f-f_{j})+v\cdot\nabla_{x}(f-f_{j})-\frac{1}{\varepsilon}\mathcal{L}(f)+\frac{1}{\varepsilon}\mathcal{L}(f_{j})|^{2}$
(4.2)

We will show that the loss term (3.1) is bounded by $O(\eta^{2})$. For the
first two terms in (4.2), it is bounded by
$\eta^{2}\,T\mu(V)\,|\Omega|\,(\varepsilon^{2}+\mu(V)^{2})$. Observe that for
any function $g$,

$\int_{-1}^{1}\left({\mathcal{L}}(g)\right)^{2}dv=-\frac{\sigma^{2}}{2}\left(\int_{-1}^{1}g(v)dv\right)^{2}+\sigma^{2}\int_{-1}^{1}g^{2}(v)dv\leq\sigma^{2}\int_{-1}^{1}g^{2}(v)dv.$
(4.3)

Since

$||f-f_{j}||_{L^{2}(\mathcal{D})}^{2}\leq||f-f_{j}||_{L^{\infty}(\mathcal{D})}^{2}\,\mu(V)<C\eta^{2},$
(4.4)

Let $g=f-f_{j}$ in (4.3) and by (4.4), we have
$\|\mathcal{L}(f)-\mathcal{L}(f_{j})\|_{L^{2}_{v}}^{2}=\|\mathcal{L}(f-f_{j})\|_{L^{2}_{v}}^{2}<O(\eta^{2})$.
So the last two terms in (4.2):

$\left\|\frac{1}{\varepsilon}\mathcal{L}(f)-\frac{1}{\varepsilon}\mathcal{L}(f_{j})\right\|_{L^{2}(\Omega\times\mathcal{D})}^{2}<O\left(\frac{\eta^{2}}{\varepsilon^{2}}\right),$
(4.5)

since $\Omega$ is bounded. Therefore, the lost term $Loss_{GE}$ in (3.1)
bounded by
$O\left(\varepsilon^{2}\eta^{2}+(\frac{\eta}{\varepsilon})^{2}\right)$.

For the inflow boundary condition, $Loss_{BC}$ is bounded by

$||f_{j}-f||_{L^{2}(\gamma_{T}^{-})}^{2}\leq
T\mu(V)|\partial\Omega|\,||f_{j}-f||_{L^{\infty}(\gamma_{T}^{-})}^{2}\leq
T\mu(V)|\partial\Omega|\,||f_{j}-f||_{L^{\infty}([0,T]\times\Omega\times\mathcal{D})}^{2}\leq
O(\eta^{2}),$

where $\gamma_{T}^{\pm}:=[0,T]\times\gamma_{\pm}$. Note that the specular
reflection boundary condition works similarly. For the initial data, denoted
by $f_{j,in}$ and $f_{in}$ respectively,

$Loss_{IC}=||f_{j,in}-f_{in}||_{L^{2}(\Omega\times\mathcal{D})}^{2}\leq||f_{j,in}-f_{in}||_{L^{\infty}(\Omega\times\mathcal{D})}^{2}\,|\Omega|\,\mu(V)\leq
O(\eta^{2}).$

Set $\eta=\eta_{j}=\frac{1}{j}$, combine all the loss terms (3.1)-(3.3), we
conclude that

$Loss_{Total}(f_{j})\leq O(\frac{1}{\varepsilon^{2}j^{2}}).$

Therefore, $Loss_{Total}(f_{j})\to 0$, as $j\to\infty$.

###  4.2 Convergence of the neural network approximated solution

In this section, we show that with the parameters
$\\{m_{[j]},w_{[j]},b_{[j]}\\}_{j=1}^{\infty}$ equipped, the neural network in
Theorem 4.1 converges to the analytic solution of the linear transport model.

######  Theorem 4.2.

Let $\\{m_{[j]},w_{[j]},b_{[j]}\\}_{j=1}^{\infty}$ be a sequence defined in
Theorem 4.1 and $f$ solves the linear transport model (2.1). As a consequence,
$Loss_{\text{Total}}(f_{j})\to 0$ implies that

$||f_{j}(\cdot,\cdot,\cdot,m_{[j]},w_{[j]},b_{[j]})-f||_{L^{\infty}([0,T];L^{2}(\Omega\times\mathcal{D}))}\to
0,$

for some finite time $t\in[0,T]$, physical variables
$(x,v)\in\Omega\times\mathcal{D}$.

Proof.  Recall the definition (4.1), then

$\left[\partial_{t}+v\cdot\nabla_{x}\right]\\{f-f_{j}\\}=d_{ge,j}(t,x,v)+\frac{1}{\varepsilon}\mathcal{L}(f)-\frac{1}{\varepsilon}\mathcal{L}(f_{j}).$
(4.6)

Define

$d_{ic,j}(x,v):=f_{0}(x,v)-f_{j}(0,x,v),$

in addition to

$d_{bc,j}(t,x,v):=g(t,x,v)-f_{j}(t,x,v)\quad\text{ at
}\,(t,x,v)\in\gamma^{-}_{T},$

for inflow boundary condition, and
$d_{bc,j}(t,x,v):=f_{j}(t,x,-v)-f_{j}(t,x,v)$ for specular reflection boundary
condition.

The $L^{2}$ norms and and inner products below stand for
$||\cdot||_{L^{2}(\Omega\times\mathcal{D})}$. Multiplying $(f-f_{j})$ onto
(4.6) and integrating over $\Omega\times\mathcal{D}$, one gets

$\displaystyle\begin{split}&\displaystyle\quad\int_{\Omega}\int_{\mathcal{D}}\varepsilon\partial_{t}(f-f_{j})^{2}\,dvdx+\int_{\gamma^{+}}(f-f_{j})^{2}\,v\cdot
n_{x}\,d\gamma-\int_{\gamma^{-}}d_{bc,j}^{2}\,|v\cdot
n_{x}|\,d\gamma\\\\[4.0pt]
&\displaystyle=\frac{2}{\varepsilon}\big{\langle}\mathcal{L}(f-f_{j}),f-f_{j}\big{\rangle}_{L^{2}}+2\big{\langle}d_{ge,j},f-f_{j}\big{\rangle}_{L^{2}}.\end{split}$
(4.7)

Recall (4.5) that $\|\mathcal{L}(f-f_{j})\|_{L^{2}}^{2}\leq C\eta^{2}$. The
right-hand-side in (4.7) is bounded by

$\frac{2}{\varepsilon}\big{\langle}\mathcal{L}(f-f_{j}),f-f_{j}\big{\rangle}_{L^{2}}\leq\frac{1}{\varepsilon}\|\mathcal{L}(f-f_{j})\|_{L^{2}}^{2}+\frac{1}{\varepsilon}\|f-f_{j}\|_{L^{2}}^{2},$

and since $\int_{\gamma^{+}}(f-f_{j})^{2}\,v\cdot n_{x}\,d\gamma\geq 0$, thus

$\varepsilon\frac{d}{dt}||f-f_{j}||_{L^{2}}^{2}\leq\underbrace{\int_{\gamma^{-}}d_{bc,j}^{2}\,|v\cdot
n_{x}|\,d\gamma+||d_{ge,j}||_{L^{2}}^{2}+\frac{C\eta^{2}}{\varepsilon}}_{:=H(t)}+(1+\frac{1}{\varepsilon})||f-f_{j}||_{L^{2}}^{2}.$

Since $\int_{\gamma^{-}}d_{bc,j}^{2}\,|v\cdot
n_{x}|\,d\gamma\leq\int_{\gamma^{-}}d_{bc,j}^{2}\,d\gamma$, and the
definitions that

$\int_{0}^{t}\int_{\gamma^{-}}\left(d_{bc,j}(s,\cdot,\cdot,\cdot)\right)^{2}\,d\gamma
ds=Loss_{BC},\quad\int_{0}^{t}||d_{ge,j}(s,\cdot,\cdot,\cdot)||_{L^{2}}^{2}\,ds=Loss_{GE},$

then

$\displaystyle\displaystyle||f-f_{j}||_{L^{2}(\Omega\times\mathcal{D})}^{2}$
$\displaystyle\leq
e^{(\frac{1}{\varepsilon}+\frac{1}{\varepsilon^{2}})t}Loss_{IC}+\frac{1}{\varepsilon}e^{(\frac{1}{\varepsilon}+\frac{1}{\varepsilon^{2}})t}\int_{0}^{t}H(s)ds$
$\displaystyle\displaystyle\lesssim
e^{(\frac{1}{\varepsilon}+\frac{1}{\varepsilon^{2}})t}\left(Loss_{IC}+\frac{1}{\varepsilon}Loss_{BC}+\frac{1}{\varepsilon}Loss_{GE}+C_{0}\,\frac{\eta^{2}}{\varepsilon^{2}}t\right)$
$\displaystyle\displaystyle\leq
e^{(\frac{1}{\varepsilon}+\frac{1}{\varepsilon^{2}})t}\left(\frac{1}{\varepsilon}Loss_{Total}+C_{0}\,\frac{\eta^{2}}{\varepsilon^{2}}t\right).$

We already know from Theorem (4.1) that $Loss_{Total}(f_{j})\leq
O(\frac{\eta^{2}}{\varepsilon^{2}})$, now take $L^{\infty}$ in $t\in[0,T]$
with $\eta=\eta_{j}=\frac{1}{j}\to 0$, then

$||f-f_{j}||_{L^{2}(\Omega\times\mathcal{D})}^{2}\leq
C^{\prime}e^{(\frac{1}{\varepsilon}+\frac{1}{\varepsilon^{2}})t}\,\eta^{2}\left(\frac{1}{\varepsilon^{3}}+C_{0}\frac{t}{\varepsilon^{2}}\right).$
(4.8)

Therefore,

$||f-f_{j}||_{L^{\infty}([0,T];L^{2}(\Omega\times\mathcal{D}))}\to
0,\quad\text{as }\,j\to\infty\,.$

######  Remark 4.3.

In [13], Liu et al. employed the hypercoercivity and Lyapunov-type functionals
to conduct sensitivity analysis for a general class of kinetic equations with
random uncertainties and multiple scales. It is possible to adopt that
framework and energy estimates analysis to study the convergence of the DNN
solution (or even improve to a stronger result of exponential decaying in
time), as presented in this section. However, due to its high complexity and
main focus of the current manuscript, we defer to a future work. Also, we
mention that a similar analysis techniques of this section can be found in
[27], where the radiative transfer model in the kinetic regime
($\varepsilon=1$), with numerical examples of different applications and its
long time behavior was studied by the DNN approach.

##  5 Numerical implementation

In this section, we are going to verify the proposed method with two examples.
The first example aims to show the accuracy of the method, and a more
challenging problem with a boundary layer is studied in the second test, which
verifies that our proposed method is able to capture the physical properties
of the model. We will compute the solutions of the model problems by using the
numerical method shown in the Appendix, then compare it with the proposed deep
learning method. As discussed before, the deep learning method is a mesh-free
method, hence inherits all the benefits of the mesh free numerical methods
meanwhile keeps a low computational cost in the testing phase. We will first
introduce the grid points of the testing data which are used in the numerical
computational. Details of the two examples will then be presented.

###  5.1 Reference solution grid points and set-up

In this section, we are going to show the computation details of the reference
solution. In order to obtain the reference solutions from the conventional
solver, we adopt the robust, implicit yet explicitly implementable Asymptotic-
Preserving (AP) numerical scheme in the even-odd decomposition framework [9],
which works with high resolution uniformly with respect to the scaling
parameter $\varepsilon$. We will use solutions of this solver as the reference
solutions to compare with the DNN approximated solutions in our numerical
tests. For the convenience of readers, we review this scheme in the Appendix.

Let $\Omega=[0,1]$. The grid points of $t$ and $x$ for the training are chosen
uniformly as follows:

$\\{(t_{i},x_{j})\\}_{i,j}\in[0,T]\times[0,1],\quad\text{ with fixed }\Delta
t,\,\Delta x.$

The integral in velocity can be computed using the quadrature rule, only few
points are needed, such as $N_{v}=32$ with $\\{v_{k}\\}_{k=1}^{N_{v}}$. Use
the grids $\\{(t=0,x_{j},v_{k})\\}_{j,k}$ for the initial condition and
$\\{(t_{i},x=0\text{ or }1,v_{k})\\}_{i,k}$ for the boundary condition. We
save the data of $f$ at fixed $t$ and $x$, for all velocity points at
$[-1,1]$, chosen uniformly with $N_{v}=32$. In the conventional finite
difference AP solver, set $x_{i}$ ($1\leq i\leq N$) and $N=40$. Here $\Delta
x=0.025$. According to the parabolic CFL of the AP scheme, $\Delta t\sim
O((\Delta x)^{2})$. In our test, $\Delta t=0.5(\Delta x)^{2}$. To summarize,
in both of our experiments, we test our models on the reference solutions
which are evaluated basing on the following fine mesh:

$\\{(t_{i},x_{j},v_{k})\\}_{i,j,k}\in[0,T]\times[0,1]\times[-1,1],\qquad\Delta
t=0.5\Delta x^{2},\,\Delta x=\frac{1}{40},\,\Delta v=2/(33-1).$

###  5.2 Test I

We consider a benchmark test for studying the linear transport model [9]. In
practice, the scaling parameter (mean free path) may differ in several orders
of magnitude from the rarefied regimes to the diffusive regimes in one
problem, thus developing methods that work uniformly with respect to this
parameter becomes important. Our learning method manages to achieve this goal,
one can test on models at any given Knudsen number without bringing additional
challenges. In this test, we consider a smooth initial condition and the
diffusive scaling. Let the initial distribution be the double-peak Maxwellian:

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\rho_{0}(x)=1+\frac{1}{2}\sin(2\pi
x),\\\\[10.0pt] \displaystyle T_{0}(x)=\frac{5+2\cos(2\pi x)}{20},\\\\[10.0pt]
\displaystyle
f(t=0,x,v)=\rho_{0}\left[\exp\left(-\left(\frac{v-0.75}{T_{0}}\right)^{2}\right)+\exp\left(-\left(\frac{v+0.75}{T_{0}}\right)^{2}\right)\right].\end{array}\right.$
(5.4)

Periodic boundary condition is considered. Assume the scattering coefficient
$\sigma=1$ and $\varepsilon=10^{-2}$.

Similar to [8], we make the data of grid points for each variable for the DNN
training. We generate the mesh uniformly with $\Delta t=0.5\Delta
x^{2},\,\Delta x=1/20,\,\Delta v=2/(33-1)$. This grid points setting provides
us with $17000$ samples associated with the the governing equation loss (3.1),
$850$ samples associated with the boundary loss (3.3) and $340$ samples
associated with the initial loss (3.2). We train the model which will be
detailed later and test the performance on a pre-generated data set which is
specified in the section (5.1).

We use a 4-layers full connected network ($3\times 256\rightarrow 256\times
256\rightarrow 256\times 256\rightarrow 256\times 1$) with bias and activated
by Tanh function. the network is trained with the Adam gradient descent with
2500 epochs. The initial learning rate is set to be 0.005 and we used a step
scheduler with step size 750 and the reduce rate 0.95.  The PINNs model is not
easy in the model training [21, 23] and there is no reliable understanding or
general method to tackle this issue [22]. One of the difficulties is the
weight tuning. In particular, when $\epsilon$ is small, the governing equation
loss (3.1) will dominate the total loss and there are multiple scales in the
loss function, bringing us an additional difficulty in the weight tuning.
Regarding this, we are going to adapt the strategy in [21, Algorithm 1]. To
make the current work more readable, we cite and present their algorithm in
the Appendix. The weights for initial and boundary losses are depicted in
Figure (1). Note that the weight for the general equation is normalized to
$1$. Our large scale experiments show that using the above-mentioned adaptive
weights gives us more accurate prediction results than the pre-set constant
weights.

Figure 1: y axis: weight; x axis: training epoch. Left: boundary loss; right:
initial condition loss. The x- axis is the training epochs and y-axis is the
weight.

We computed the relative error of (6.3), more precisely:

$\text{relative
error}=\frac{\|u_{\text{True}}-u_{\text{Predicted}}\|}{\|u_{\text{True}}\|},$

where $u_{\text{True}}=\int_{-1}^{+1}fdv$ and we denote the true solution as
$f$. The evolution of the relative errors with respected to the training epoch
is shown in Figure (2).

Figure 2: Relative loss defined by (5.2), with respect to the number of
training epochs. y axis: relative loss as defined in (5.2); x axis: training
epoch.

Note that the largest relative error stabilizes at less than 10 percents when
the neural network converges and the number of epochs reach about $2000$. We
also plot the (6.3) of the true and predicted solution in Figure (3) at
different time.

Figure 3: Plot of the density at different output time. The blue curve is the
NN predicted solution and the red curve is the reference solution.

We can observe from the Figure (3) that the proposed neural network method
gives an accurate approximation to the reference solution; this approximation
is close even when time is large. Also, periodic boundary conditions are
satisfied. This example verifies that our method is accurate and then we can
benefit from the many advantages of the learning algorithms. One of the
advantages is the high efficiency in predicting new samples once the model is
trained. This is very useful since our framework is mesh-less and compare to
the conventional numerical methods, we are able to calculate the solution at
any given points in the support of the equation. One hence can study the
physics of the model efficiently.

###  5.3 Test II

In this experiment, we consider another benchmark test with the incoming
boundary data [9]. This problem is more complicated and challenging, since the
solution contains the boundary layer, and we manage to see the DNN approach
can capture the solution behavior especially near the boundary. The initial
condition is given by $f(0,x,v)=0$, and boundary conditions are

$f(t,0,v)=1,\quad v\geq 0;\quad f(t,1,v)=0,\quad v\leq 0.$

Consider the diffusive regime with $\varepsilon=10^{-3}$, and $\sigma=1$. Set
$\Delta t=0.5\cdot\Delta x^{2}$ and $\Delta x=1/25$. This gives us 41225
samples for the governing equation loss, 1746 samples for the boundary loss
and 425 for the initial condition loss. We use a 4-layers fully connected
network ($3\times 256\rightarrow 256\times 256\rightarrow 256\times
256\rightarrow 256\times 1$) activated by Tanh and trained by the Adam
gradient descent for 400 epochs. The initial learning rate is set to be 0.0005
and a step schedule is used which will reduced the rate by 5 percents every 50
epochs. Same as the first example, we will adopt the adaptive weight balance
strategy. The weight evolution of the initial and boundary loss is shown in
Figure (4).

Figure 4: y axis: weight; x axis: training epoch. Right: boundary loss; left:
initial condition loss. The loss for the governing equation is normalized to
1.

We finally present the results of the experiment in figure 5.

Figure 5: y axis: can be calculated by (6.3); x axis: points in spatial
direction x. Each figure is the result at a time step, please see the
subtitles for the details of the time level. The blue curve is the predicted
solution and the red curve is the reference solution.

The most challenging property of this example is the existing of the boundary
layer. We can see from the Figure (5), the proposed method is able to capture
the boundary layer. This is the consequence of minimizing the combined loss
and the solution hence has the all the physical properties of the equation.

##  6 Conclusion and future work

In this work, we proposed to solve the linear transport model by a learning
method.  We consider the diffusive scaling with a small Knudsen number in our
numerical tests, while our analysis applies to all orders of the Knudsen
number. The asymptotic-preserving solver [9], as a robust traditional
numerical method, was designed to tackle the stiffness of the model brought by
the small relaxing parameter, without resolving the numerical discretizations.
However, it requires a solid understanding of kinetic theory and not very
feasible to practical applications in physics or engineering. Our learning
method, on the other hand, is mesh-free, easy to implement (no matter how
complicated the initial or boundary conditions are), provides the numerical
solution at any given points, while does not need a strong background on
kinetic theory, thus making it more applicable to general research fields.
Theoretically, we prove that the total loss function vanishes as the network
converges, then show that the neural network solution converges to the
analytic solution pointwisely. In order to demonstrate the advantages of the
learning method, we test on two benchmark examples, whose results show that
our method is accurate and can capture the quantities of interests accurately,
given challenging initial or boundary conditions.

In the future, we will extend our proposed method to high-dimensional kinetic
problems with uncertainties, and develop new training methods–in particular
work on weight balancing of different loss terms. We may also consider
applying the PINN framework to solve inverse problems associated with kinetic
models.

Appendix

A. The asymptotic-preserving method  We briefly recall [9] for the
reformulation to diffusive relaxation system of the linear transport equation
(2.1), and its diffusion limit system as $\varepsilon\to 0$. This also
prepares us to study the asymptotic behavior of the distribution function,
which will be studied in a follow-up work.

First, we split (2.1) into two equations for $v>0$:

$\begin{split}\displaystyle&\varepsilon\partial_{t}f(v)+v\partial_{x}f(v)=\frac{\sigma(x)}{\varepsilon}\left(\frac{1}{2}\int_{-1}^{1}f(v)\,dv-f(v)\right),\\\\[6.0pt]
\displaystyle&\varepsilon\partial_{t}f(-v)-v\partial_{x}f(-v)=\frac{\sigma(x)}{\varepsilon}\left(\frac{1}{2}\int_{-1}^{1}f(v)\,dv-f(-v)\right),\end{split}$
(6.1)

In this case consider the even and odd parities

$\begin{split}\displaystyle
r(t,x,v)&=\frac{1}{2}[f(t,x,v)+f(t,x,-v)],\\\\[4.0pt] \displaystyle
j(t,x,v)&=\frac{1}{2\varepsilon}[f(t,x,v)-f(t,x,-v)].\end{split}$

Adding and subtracting the two equations in (6.1) leads to

$\left\\{\begin{split}\displaystyle&\partial_{t}r+v\partial_{x}j=\frac{\sigma(x)}{\varepsilon^{2}}(\rho-r),\\\\[4.0pt]
\displaystyle&\partial_{t}j+\frac{v}{\varepsilon^{2}}\partial_{x}r=-\frac{\sigma(x)}{\epsilon^{2}}j.\end{split}\right.$
(6.2)

where

$\rho(t,x)=\int_{0}^{1}r\,dv.$ (6.3)

As $\varepsilon\rightarrow 0^{+}$, (6.2) yields

$r=\rho,\qquad j=-\frac{v}{\sigma(x)}\partial_{x}\rho.$

Substituting this into the first equation of (6.2) and integrating over $v$,
one gets the limiting diffusion equation [1]:

$\left\\{\begin{split}\displaystyle&j=-\frac{v}{\sigma(x)}\partial_{x}\rho,\\\\[4.0pt]
\displaystyle&\partial_{t}\rho=\partial_{x}\left(\frac{1}{3\sigma(x)}\partial_{x}\rho\right).\end{split}\right.$
(6.4)

We solve the diffusive relaxation system (6.2) by splitting it into a
relaxation step, followed by a transport step. One can check details of the
discretized scheme in [9], we omit it here.

B. Weights Balance Algorithm  We review the weight balance algorithm studied
in [21], which designs appropriate weights of different loss terms in the
total loss function.

1 Consider a physics-informed neural network $f_{\theta}(x)$ with parameters
$\theta$ and a loss function
$\mathcal{L}=\mathcal{L}_{G}+\sum_{i=1}^{M}\lambda_{i}\mathcal{L}_{i}(\theta).$
where $\mathcal{L}_{G}$ is the governing equation loss and $\mathcal{L}_{i}$
are the other losses (initial condition and etc.). $\lambda_{i}$ are the
weights to balance the interplay of the losses.;

2 for  _$n=1,...,S$_ do

3 Compute $\hat{\lambda}_{i}$ by:
$\hat{\lambda}_{i}=\frac{\max_{\theta}|\nabla_{\theta}\mathcal{L}_{G}(\theta_{n})|}{\overline{|\nabla_{\theta}\mathcal{L}_{i}(\theta_{n})|}},$
where $\overline{|\nabla_{\theta}\mathcal{L}_{i}(\theta_{n})|}$ is the mean of
$|\nabla_{\theta}\mathcal{L}_{i}(\theta_{n})|$ with respected to $\theta_{n}$;

4 Update the weights $\lambda_{i}$ using a moving average:
$\lambda_{i}=(1-\alpha)\lambda_{i}+\alpha\lambda_{i},i=1,...M,$ where $\alpha$
is a constant and the authors suggest that $\alpha=0.9$;

5 Update the parameter $\theta$ via the gradient descent;

6

7 end for

Algorithm 1 Learning rate annealing for the PINN [21]

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