# Constraint-Aware Neural Networks for Riemann Problems

arXiv: https://arxiv.org/abs/1904.12794 · HTML: https://mlsanity.com/paper/1904.12794/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2019.

## Built on by (most cited first, 14 of 14)

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- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)
- [Addressing Discontinuous Root-Finding for Subsequent Differentiability in Machine Learning, Inverse Problems, and Control](https://mlsanity.com/paper/2306.12413/) (2023)
- [Neural network-based Godunov corrections for approximate Riemann solvers using bi-fidelity learning](https://mlsanity.com/paper/2503.13248/) (2025)
- [Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification](https://mlsanity.com/paper/2505.04263/) (2025)

## Abstract (arXiv)

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.