Coupled Integral PINN for Discontinuity

TL;DR

The paper tackles forward solving of nonlinear hyperbolic conservation laws with shocks using physics-informed neural networks, where standard strong-form PINNs fail to recover physically admissible entropy solutions. It introduces the Coupled Integral PINN (CI-PINN), a mesh-free dual-network framework that learns a primitive state $u(x,t)$ and an integral potential $S(x,t)$ with constraints $q \approx div S$ and $d_t S + F(q) \approx 0$, along with entropy admissibility. The authors provide optimization-theory insights showing barriers for sharp shocks and demonstrate that CI-PINN delivers sharper shocks, correct rarefaction behavior, and robust performance on 1D benchmarks (Burgers, Buckley–Leverett, Euler) and 2D problems (Euler, SWE), outperforming strong-form PINNs and several integral-form baselines. This approach offers a flexible, mesh-free solver for discontinuous PDEs, maintaining conservation laws without flux reconstruction and enabling integration with existing PINN enhancements.

Abstract

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.

Figures (16)

• Figure 1: CI-PINN architecture to solve the forward problem of nonlinear PDEs.
• Figure 2: We utilize a square-wave initial condition ($u(x,0)=1$ for $x\in(-0.5,0)$ as in the left-most panels), which naturally induces both a shock (where characteristics pile up, ie., sudden changes in right-most panels) and a rarefaction wave (where characteristics fan apart, ie., linear slope in right-most panels.)
• Figure 3: In these shock-tube profiles, the contact discontinuity appears as a sharp change in density while the velocity and pressure remain essentially unchanged across the interface. The plateau states are the flat segments between the smooth expansion region and the sharp jumps, and their constant levels are fixed by conservation across the discontinuities (Rankine--Hugoniot) together with the expansion relations.
• Figure 4: Selective component on 2D system, full result please refer to appendix \ref{['Appendix:2dexp']}
• Figure A1: Comparison of predictions for Burger problem.

Theorems & Definitions (8)

• Proposition 2.1: Spurious smooth low-loss solutions dominate the strong-form objective
• Remark 2.2
• Proposition 2.3: Strong-form blow-up as a discontinuity sharpens
• proof
• Corollary 2.4: The Optimization barrier
• Remark 2.5
• Remark 2.6: How CI-PINN avoids strong-form shock blow-up
• proof : Proof (normal-line mean-value argument)