Solving the Boltzmann Equation with a Neural Sparse Representation

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.