# Certified machine learning: A posteriori error estimation for physics-informed neural networks

arXiv: https://arxiv.org/abs/2203.17055 · HTML: https://mlsanity.com/paper/2203.17055/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 31 Mar 2022.

## Builds on (most cited first, 5 of 5)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators](https://mlsanity.com/paper/2109.10793/) (2021)
- [Reliable Prediction Errors for Deep Neural Networks Using Test-Time Dropout](https://mlsanity.com/paper/1904.06330/) (2019)
- [Surrogate-data-enriched Physics-Aware Neural Networks](https://mlsanity.com/paper/2112.05489/) (2021)

## Built on by (most cited first, 15 of 15)

- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [Efficient Error Certification for Physics-Informed Neural Networks](https://mlsanity.com/paper/2305.10157/) (2023)
- [Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs](https://mlsanity.com/paper/2210.03426/) (2022)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [Prediction error certification for PINNs: Theory, computation, and application to Stokes flow](https://mlsanity.com/paper/2508.07994/) (2025)
- [Error estimation for physics-informed neural networks with implicit Runge-Kutta methods](https://mlsanity.com/paper/2401.05211/) (2024)
- [Astral: training physics-informed neural networks with error majorants](https://mlsanity.com/paper/2406.02645/) (2024)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)
- [Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks](https://mlsanity.com/paper/2601.19818/) (2026)
- [DeepSPoC: A Deep Learning Based Sequential Propagation of Chaos](https://mlsanity.com/paper/2408.16403/) (2024)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Certified and accurate computation of function space norms of deep neural networks](https://mlsanity.com/paper/2603.06431/) (2026)
- [Building Trust in PINNs: Error Estimation through Finite Difference Methods](https://mlsanity.com/paper/2603.15526/) (2026)
- [Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds](https://mlsanity.com/paper/2606.12050/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.