# Coupled Integral PINN for Discontinuity

arXiv: https://arxiv.org/abs/2411.11276 · HTML: https://mlsanity.com/paper/2411.11276/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 37 references that are in the map.
- First posted on arXiv 18 Nov 2024.

## Builds on (most cited first, 12 of 12)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Random Weight Factorization Improves the Training of Continuous Neural Representations](https://mlsanity.com/paper/2210.01274/) (2022)
- [PINNs for the Solution of the Hyperbolic Buckley-Leverett Problem with a Non-convex Flux Function](https://mlsanity.com/paper/2112.14826/) (2021)

## Built on by (most cited first, 1 of 1)

- [Enhancing PINN Accuracy for the RLW Equation: Adaptive and Conservative Approaches](https://mlsanity.com/paper/2511.11638/) (2025)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) solve forward PDEs by minimizing residual losses from the governing equations with initial and boundary conditions, but they often struggle with discontinuities such as shocks. In contrast, finite volume methods (FVM) handle discontinuities by enforcing integral conservation, which admits weak solutions. Motivated by this, we propose a Coupled Integral PINN (CI-PINN) that augments a standard PINN with an auxiliary network for integral potentials and coupled integral constraints. This improves robustness near shocks while avoiding meshing and the numerical flux integration/reconstruction used in classical schemes. We validate CI-PINN on forward benchmarks including Burgers, Buckley--Leverett, the Euler system, and the Shallow-Water equations.