# Efficient Error Certification for Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2305.10157 · HTML: https://mlsanity.com/paper/2305.10157/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 47 references that are in the map.
- First posted on arXiv 17 May 2023.

## Builds on (most cited first, 27 of 27)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers](https://mlsanity.com/paper/2011.13824/) (2020)
- [An approach to reachability analysis for feed-forward ReLU neural networks](https://mlsanity.com/paper/1706.07351/) (2017)
- [A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks](https://mlsanity.com/paper/1902.08722/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Bound](https://mlsanity.com/paper/2205.00263/) (2022)
- [Learning to Branch](https://mlsanity.com/paper/1803.10150/) (2018)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Robustness Verification for Transformers](https://mlsanity.com/paper/2002.06622/) (2020)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Improved Branch and Bound for Neural Network Verification via Lagrangian Decomposition](https://mlsanity.com/paper/2104.06718/) (2021)
- [Certified machine learning: A posteriori error estimation for physics-informed neural networks](https://mlsanity.com/paper/2203.17055/) (2022)
- [MixTrain: Scalable Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1811.02625/) (2018)
- [PaRoT: A Practical Framework for Robust Deep Neural Network Training](https://mlsanity.com/paper/2001.02152/) (2020)

## Built on by (most cited first, 10 of 10)

- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Error estimation for physics-informed neural networks with implicit Runge-Kutta methods](https://mlsanity.com/paper/2401.05211/) (2024)
- [Scalable Verification of Neural Control Barrier Functions Using Linear Bound Propagation](https://mlsanity.com/paper/2511.06341/) (2025)
- [PINNfluence: Interpreting PINNs through Influence Functions](https://mlsanity.com/paper/2409.08958/) (2024)
- [Generalization vs. Memorization in Autoregressive Deep Learning: Or, Examining Temporal Decay of Gradient Coherence](https://mlsanity.com/paper/2509.00024/) (2025)
- [LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries](https://mlsanity.com/paper/2510.25731/) (2025)
- [BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations](https://mlsanity.com/paper/2602.14853/) (2026)
- [Building Trust in PINNs: Error Estimation through Finite Difference Methods](https://mlsanity.com/paper/2603.15526/) (2026)
- [Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees](https://mlsanity.com/paper/2603.19165/) (2026)
- [Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds](https://mlsanity.com/paper/2606.12050/) (2026)

## Abstract (arXiv)

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain - a measure akin to the tolerance of numerical solvers - focusing instead on point-wise comparisons between their solution and the ones obtained by a solver on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce partial-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs - Burgers' and Schr"odinger's equations -, and two more challenging ones with real-world applications - the Allan-Cahn and Diffusion-Sorption equations.