Data-driven, structure-preserving approximations to entropy-based moment closures for kinetic equations. (English) Zbl 1516.65112
Summary: We present a data-driven approach for approximating entropy-based closures of moment systems from kinetic equations. The proposed closure learns the entropy function by fitting the map between the moments and the entropy of the moment system, and thus does not depend on the spacetime discretization of the moment system or specific problem configurations such as initial and boundary conditions. With convex and \(C^2\) approximations, this data-driven closure inherits several structural properties from entropy-based closures, such as entropy dissipation, hyperbolicity, and H-Theorem. We construct convex approximations to the Maxwell-Boltzmann entropy using convex splines and neural networks, test them on the plane source benchmark problem for linear transport in slab geometry, and compare the results to the standard, entropy-based systems which solve a convex optimization problem to find the closure. Numerical results indicate that these data-driven closures provide accurate solutions in much less computation time than that required by the optimization routine.
MSC:
65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
35Q20 | Boltzmann equations |
65K10 | Numerical optimization and variational techniques |
82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
82C70 | Transport processes in time-dependent statistical mechanics |