Separable Physics-informed Neural Networks for Solving the BGK Model of
  the Boltzmann Equation
Jaemin Oh Seung Yeon Cho Seok-Bae Yun Eunbyung Park Youngjoon Hong
In this study, we introduce a method based on Separable Physics-Informed
Neural Networks (SPINNs) for effectively solving the BGK model of the Boltzmann
equation. While the mesh-free nature of PINNs offers significant advantages in
handling high-dimensional partial differential equations (PDEs), challenges
arise when applying quadrature rules for accurate integral evaluation in the
BGK operator, which can compromise the mesh-free benefit and increase
computational costs. To address this, we leverage the canonical polyadic
decomposition structure of SPINNs and the linear nature of moment calculation,
achieving a substantial reduction in computational expense for quadrature rule
application. The multi-scale nature of the particle density function poses
difficulties in precisely approximating macroscopic moments using neural
networks. To improve SPINN training, we introduce the integration of Gaussian
functions into SPINNs, coupled with a relative loss approach. This modification
enables SPINNs to decay as rapidly as Maxwellian distributions, thereby
enhancing the accuracy of macroscopic moment approximations. The relative loss
design further ensures that both large and small-scale features are effectively
captured by the SPINNs. The efficacy of our approach is demonstrated through a
series of five numerical experiments, including the solution to a challenging
3D Riemann problem. These results highlight the potential of our novel method
in efficiently and accurately addressing complex challenges in computational
physics.