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Efficient Error Certification for Physics-Informed Neural Networks
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Introduces ∂-CROWN, a post-training framework that certifies worst-case PDE residual errors of PINNs over their continuous domain using closed-form linear bounds on derivatives plus greedy input branching.
Problem
PINNs are evaluated only on finite point sets, so no guarantees exist on worst-case residual or boundary/initial condition errors across the continuous spatio-temporal domain.
Approach
Define globally correct PINN conditions on residual, initial and boundary errors, and extend CROWN/alpha-CROWN to bound partial derivatives of a fully connected network in closed form (hybrid backward/forward substitution), combined with McCormick envelopes for the nonlinear PDE residual and a greedy input-branching refinement.
Key results
- On Diffusion-Sorption, 10^4 Monte Carlo samples give residual max 1.10e-3 while 10^6 samples give 21.09; the certified upper bound is 21.34.
- On Burgers' equation, ∂-CROWN certifies residual |f|^2 <= 1.03e-1 versus empirical 1.80e-2 with 10^6 samples, with about 2.8e5 s runtime.
- In a fixed-time comparison on Burgers' residual, ∂-CROWN gives 1.30e1 versus LiRPA 1.78e2 and IBP 2.78e3.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ∂-CROWN instead of IBP for residual certification | Burgers PINN · Burgers residual | certified max residual squared: 2.78×10^3 -> 1.30 \times 10^1 | Helped |
| Use ∂-CROWN instead of LiRPA for residual certification | Burgers PINN · Burgers residual | certified max residual squared: 1.78 \times 10^2 -> 1.30 \times 10^1 | Helped |
| Use LiRPA instead of ∂-CROWN for initial-condition certification | Burgers PINN · Burgers initial condition | certified max initial-condition error squared: 2.63\times 10^{-6} -> 2.23\times 10^{-6} | Helped |
| Use LiRPA instead of ∂-CROWN for boundary certification at x = -1 | Burgers PINN · Burgers boundary, x = -1 | certified max boundary error squared: 6.63\times 10^{-7} -> 6.34\times 10^{-7} | Helped |
| Use LiRPA instead of ∂-CROWN for boundary certification at x = 1 | Burgers PINN · Burgers boundary, x = 1 | certified max boundary error squared: 9.39\times 10^{-7} -> 9.12\times 10^{-7} | Helped |
| Train with physics-informed adversarial training instead of standard training | Burgers PINN · Burgers residual | certified max residual squared: 1.03×10^{-1} -> 2.39\cdot 10^{-2} | Helped |
| Train with physics-informed adversarial training instead of standard training | Burgers PINN · Burgers residual | Monte Carlo max residual squared: 1.80×10^{-2} -> 3.60\cdot10^{-3} | Helped |
| Increase the number of greedy input branches (ablation) | ∂-CROWN · Burgers and Schrödinger residual | certified residual bound | Helped |
Surprising findings
- Monte Carlo estimates of the maximum residual for Diffusion-Sorption change by four orders of magnitude between 10^4 and 10^6 samples, showing sampling-based testing can be misleading.
Limitations
- Residual verification runtime is high due to the many branchings needed for tight bounds.
- Bounds loosen for higher-order PDEs and deeper solution networks, degrading tightness of incomplete verifiers.
- Method is restricted to fully connected networks and was applied only to low-dimensional PINNs with tanh activations.
Experimental setup
Datasets: numerical finite-difference solver solutions for Burgers' equation
Benchmarks: Burgers' equation PINN Schrödinger equation PINN Allen-Cahn equation PINN Diffusion-Sorption equation PINN
Metrics: certified upper bound on squared residual/condition error Monte Carlo empirical lower bound runtime (s)
Baselines: Interval Bound Propagation LiRPA Monte Carlo sampling
Builds on: CROWN alpha-CROWN McCormick envelopes PINN
Closest papers by meaning
- Competitive Physics Informed Networks (ICLR 2023 · similarity 0.92)
- Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.91)
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.90)
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.89)
- PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks (ICLR 2024 · similarity 0.89)
- S$^{2}$-PINN: Stochastic Separable Physics-Informed Neural Networks (NeurIPS 2026 · similarity 0.89)
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