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.

翻译：本研究提出了一种基于可分离物理信息神经网络（SPINNs）的方法，用于高效求解玻尔兹曼方程的BGK模型。尽管PINNs的无网格特性在处理高维偏微分方程（PDEs）时具有显著优势，但在BGK算子中应用求积法则进行精确积分计算时会面临挑战，这可能会削弱无网格优势并增加计算成本。为解决这一问题，我们利用SPINNs的典型多线性分解结构以及矩计算的线性特性，大幅降低了求积法则应用中的计算开销。粒子密度函数的多尺度特性给神经网络精确逼近宏观矩带来了困难。为提升SPINN训练效果，我们引入高斯函数与SPINNs的融合策略，并采用相对损失函数。这一改进使得SPINNs能够像麦克斯韦分布一样快速衰减，从而提高了宏观矩逼近的精度。相对损失设计进一步确保了SPINNs能够有效捕捉大尺度与小尺度特征。通过一系列包含五个数值实验（其中包括求解具有挑战性的三维黎曼问题）的验证，证明了该方法的有效性。这些结果突显了本方法在高效、精确解决计算物理中复杂问题方面的潜力。