SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page C186-C215, April 2024.
Abstract. We consider the neural sparse representation to solve the Boltzmann equation with BGK and quadratic collision models, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and physical space so as to avoid the discretization in space and time. Different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision models, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All of these will significantly improve the efficiency of the network when solving the Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multiscale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod tube problems and a 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods.

中文翻译：

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用神经稀疏表示求解玻尔兹曼方程

SIAM 科学计算杂志，第 46 卷，第 2 期，C186-C215 页，2024 年 4 月。
摘要。我们考虑使用神经稀疏表示来求解带有 BGK 和二次碰撞模型的玻尔兹曼方程，其中提出了一种基于网络的 ansatz，可以以极高的效率逼近分布函数。准确地说，在时间和物理空间上采用全连接的神经网络，以避免空间和时间上的离散化。BGK 和二次碰撞模型的微观速度采用了不同的低秩表示，导致自由度显着降低。我们使用正则多元分解来近似 BGK 模型中的离散速度分布。对于二次碰撞模型，在 BGK 解的基础上构建了数据驱动的、基于 SVD 的线性基础。所有这些都将显着提高网络求解玻尔兹曼方程时的效率。此外，还提出了专门设计的自适应权重损失函数，并采用多尺度输入和麦克斯韦分裂策略，进一步提高逼近效率并加快学习过程。几个数值实验，包括一维波和 Sod 管问题以及二维波问题，证明了这些神经稀疏表示方法的有效性。