Solving Boltzmann equation with neural sparse representation

Abstract: We consider the neural sparse representation to solve Boltzmann equation with BGK and quadratic collision model, 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 spatial space so as to avoid the discretization in space and time. The different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision model, 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 these will significantly improve the efficiency of the network when solving Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multi-scale 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 problems and 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods.