MATHICSE Technical Report: Constraint-Aware Neural Networks for Riemann Problems

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satised to obtain reliable results. However, standard machine learning algorithms are generally not tailored to respect such constraints. We propose two different strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible uid ow. Precisely, in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic ux model problem, an isothermal two-phase ow model, and the Euler equations. We demonstrate that a decrease in the constraint deviation correlates with low discretization errors for all model problems, in addition to the structural advantage of fullfilling the constraint.