  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.
