# Residual-based error bound for physics-informed neural networks

arXiv: https://arxiv.org/abs/2306.03786 · HTML: https://mlsanity.com/paper/2306.03786/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2023.

## Built on by (most cited first, 9 of 9)

- [One-Shot Transfer Learning for Nonlinear ODEs](https://mlsanity.com/paper/2311.14931/) (2023)
- [Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles](https://mlsanity.com/paper/2505.06459/) (2025)
- [Exact and approximate error bounds for physics-informed neural networks](https://mlsanity.com/paper/2411.13848/) (2024)
- [Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks](https://mlsanity.com/paper/2601.19818/) (2026)
- [Augmented neural forms with parametric boundary-matching operators for solving ordinary differential equations](https://mlsanity.com/paper/2404.19454/) (2024)
- [Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains](https://mlsanity.com/paper/2510.22497/) (2025)
- [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)
- [Physics-Informed Neural Network Surrogate for Oxygen Vacancy Dynamics in epitaxial SrTiO3 on Si memristors via Dynamic Spectral Optimization](https://mlsanity.com/paper/2609.02966/) (2026)

## Abstract (arXiv)

Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously evaluate the error bound of Physics-Informed Neural Networks (PINNs) on most linear ordinary differential equations (ODEs), certain nonlinear ODEs, and first-order linear partial differential equations (PDEs). The error bound is based purely on equation structure and residual information and does not depend on assumptions of how well the networks are trained. We propose algorithms that bound the error efficiently. Some proposed algorithms provide tighter bounds than others at the cost of longer run time.