# Constraint-Aware Neural Networks for Riemann Problems

## TL;DR

This paper introduces two strategies for developing neural networks that inherently respect physical constraints, improving the accuracy and reliability of simulations involving nonlinear wave motion in fluid dynamics.

## Contribution

The paper proposes novel methods for creating constraint-aware neural networks tailored for Riemann problems in fluid flow simulations.

## Key findings

- Constraint-aware networks reduce deviation from physical laws.
- Lower constraint deviation correlates with decreased discretization errors.
- Methods improve reliability of wave front predictions in simulations.

## Abstract

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 satisfied 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 fluid flow. 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 flux model problem, an isothermal two-phase flow 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 fulfilling the constraint.