# A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I

arXiv: https://arxiv.org/abs/2307.04121 · HTML: https://mlsanity.com/paper/2307.04121/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 50 references that are in the map.
- First posted on arXiv 9 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Physics-informed neural networks for modeling rate- and temperature-dependent plasticity](https://mlsanity.com/paper/2201.08363/) (2022)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [PhySRNet: Physics informed super-resolution network for application in computational solid mechanics](https://mlsanity.com/paper/2206.15457/) (2022)
- [Machine Learning-Accelerated Computational Solid Mechanics: Application to Linear Elasticity](https://mlsanity.com/paper/2112.08676/) (2021)
- [Spatio-Temporal Super-Resolution of Dynamical Systems using Physics-Informed Deep-Learning](https://mlsanity.com/paper/2212.04457/) (2022)
- [RoeNets: Predicting Discontinuity of Hyperbolic Systems from Continuous Data](https://mlsanity.com/paper/2006.04180/) (2020)

## Abstract (arXiv)

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.