# Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

arXiv: https://arxiv.org/abs/2508.07994 · HTML: https://mlsanity.com/paper/2508.07994/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 51 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 14 of 14)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [Certified machine learning: A posteriori error estimation for physics-informed neural networks](https://mlsanity.com/paper/2203.17055/) (2022)
- [Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs](https://mlsanity.com/paper/2210.03426/) (2022)
- [An Approximation Theory Perspective on Machine Learning](https://mlsanity.com/paper/2506.02168/) (2025)
- [Infinite-dimensional input-to-state stability and Orlicz spaces](https://mlsanity.com/paper/1609.09741/) (2016)
- [PINNs error estimates for nonlinear equations in ℝ-smooth Banach spaces](https://mlsanity.com/paper/2305.11915/) (2023)

## Built on by (most cited first, 4 of 4)

- [A Hybrid Framework for Healing Semigroups with Machine Learning](https://mlsanity.com/paper/2509.01763/) (2025)
- [LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries](https://mlsanity.com/paper/2510.25731/) (2025)
- [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)

Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics-informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup-based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples - due to the need to compute quantities related to input-to-state stability - we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.