# Separable Physics-informed Neural Networks for Solving the BGK Model of the Boltzmann Equation

arXiv: https://arxiv.org/abs/2403.06342 · HTML: https://mlsanity.com/paper/2403.06342/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 56 references that are in the map.
- First posted on arXiv 10 Mar 2024.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equation](https://mlsanity.com/paper/2207.01765/) (2022)
- [Randomized Forward Mode of Automatic Differentiation For Optimization Algorithms](https://mlsanity.com/paper/2310.14168/) (2023)

## Built on by (most cited first, 8 of 8)

- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [Separable Operator Networks](https://mlsanity.com/paper/2407.11253/) (2024)
- [FourierSpecNet: Neural Collision Operator Approximation Inspired by the Fourier Spectral Method for Solving the Boltzmann Equation](https://mlsanity.com/paper/2504.20408/) (2025)
- [Solving Euler equations with Multiple Discontinuities via Separation-Transfer Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.20361/) (2025)
- [Micro-Macro Tensor Neural Surrogates for Uncertainty Quantification in Collisional Plasma](https://mlsanity.com/paper/2512.24205/) (2025)
- [Learning collision operators from plasma phase space data using differentiable simulators](https://mlsanity.com/paper/2601.10885/) (2026)
- [A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networks](https://mlsanity.com/paper/2604.04971/) (2026)
- [Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes](https://mlsanity.com/paper/2604.24745/) (2026)

## Abstract (arXiv)

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.