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Coupled Integral PINN for Discontinuity

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed neural network training · 1 citations (OpenAlex)

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Proposes CI-PINN, a dual-network PINN (primitive net plus potential net for integral potentials) with coupled integral, entropy, and adaptive strong-form losses to solve forward hyperbolic conservation laws with shocks.

Problem

Vanilla PINNs struggle to capture shocks and discontinuities in forward hyperbolic PDEs because strong-form residuals are ill-posed and converge to smeared solutions.

Approach

Enforce conservation laws in integral form via an auxiliary potential network S whose divergence matches the conserved state (q ≈ ∇·S) and whose time derivative balances fluxes (∂_t S_j + F_j(q) ≈ 0), plus an entropy admissibility penalty and a shock-masked strong-form loss, all mesh-free.</undefined>

Key results

• On Burgers, CI-PINN global L2 error 47.04e-3 vs 96.15e-3 for the next best (cvPINN).
• On 2D SWE, CI-PINN reduces L2 error for height h from 225.81e-3 (cvPINN) to 26.10e-3 (an 88.4% reduction).
• On the Lax shock tube, CI-PINN global L1 error is 14.83e-3 vs 142.00e-3 for cvPINN.

Surprising findings

• Integral-form methods (cvPINN, IPINN) can be less stable than other baselines in smooth rarefaction regions, while CI-PINN is more balanced.
• CI-PINN's L_infinity error is not uniformly best; on the BL and some Euler cases other methods (e.g., AdaLR, Modified MLP, IPINN) achieve better L_infinity.

Limitations

• Relies on loss weighting hyperparameters and penalty tuning, which the authors note as a sensitivity issue.
• Theoretical propositions are sketched with informal scaling arguments and 'generic' assumptions.
• Some tables show CI-PINN with the best L2/L1 but not L_infinity, and rarefaction-region results are mixed on Burgers.

Experimental setup

Datasets: Burgers equation Buckley-Leverett Sod shock tube Lax shock tube 2D Euler Riemann problem 2D shallow-water equations

Benchmarks: Burgers Buckley-Leverett Sod shock tube Lax shock tube 2D Euler Riemann 2D SWE

Metrics: L1 L2 L_infinity error (global, shock, rarefaction regions)

Baselines: cvPINN IPINN Modified MLP Causal training Random Weight Factorization Adaptive learning rate Vanilla PINN ProbConserv

Builds on: PINN cvPINN IPINN Causal training Random Weight Factorization

Closest papers by meaning

• PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks (ICLR 2024 · similarity 0.86)
• RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.86)
• Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.86)
• Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.86)
• PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.85)
• PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling (NeurIPS 2025 · similarity 0.85)