# Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs

arXiv: https://arxiv.org/abs/2210.03426 · HTML: https://mlsanity.com/paper/2210.03426/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 50 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 9 of 9)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness](https://mlsanity.com/paper/1905.11427/) (2019)
- [Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators](https://mlsanity.com/paper/2109.10793/) (2021)
- [Certified machine learning: A posteriori error estimation for physics-informed neural networks](https://mlsanity.com/paper/2203.17055/) (2022)
- [Infinite-dimensional input-to-state stability and Orlicz spaces](https://mlsanity.com/paper/1609.09741/) (2016)

## Built on by (most cited first, 8 of 8)

- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [Prediction error certification for PINNs: Theory, computation, and application to Stokes flow](https://mlsanity.com/paper/2508.07994/) (2025)
- [PINNs error estimates for nonlinear equations in ℝ-smooth Banach spaces](https://mlsanity.com/paper/2305.11915/) (2023)
- [PINNfluence: Interpreting PINNs through Influence Functions](https://mlsanity.com/paper/2409.08958/) (2024)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)
- [Building Trust in PINNs: Error Estimation through Finite Difference Methods](https://mlsanity.com/paper/2603.15526/) (2026)
- [Man, Machine, and Mathematics](https://mlsanity.com/paper/2604.27052/) (2026)
- [One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs](https://mlsanity.com/paper/2608.22026/) (2026)

## Abstract (arXiv)

Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.