# Error estimation for physics-informed neural networks with implicit Runge-Kutta methods

arXiv: https://arxiv.org/abs/2401.05211 · HTML: https://mlsanity.com/paper/2401.05211/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 23 references that are in the map.
- First posted on arXiv 10 Jan 2024.

## Builds on (most cited first, 10 of 10)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networks](https://mlsanity.com/paper/2109.04304/) (2021)
- [Certified machine learning: A posteriori error estimation for physics-informed neural networks](https://mlsanity.com/paper/2203.17055/) (2022)
- [Efficient Error Certification for Physics-Informed Neural Networks](https://mlsanity.com/paper/2305.10157/) (2023)
- [PINNSim: A Simulator for Power System Dynamics based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2303.10256/) (2023)
- [A Frequency Domain Approach to Predict Power System Transients](https://mlsanity.com/paper/2111.01103/) (2021)

## Built on by (most cited first, 2 of 2)

- [Exact and approximate error bounds for physics-informed neural networks](https://mlsanity.com/paper/2411.13848/) (2024)
- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)

## Abstract (arXiv)

The ability to accurately approximate trajectories of dynamical systems enables their analysis, prediction, and control. Neural network (NN)-based approximations have attracted significant interest due to fast evaluation with good accuracy over long integration time steps. In contrast to established numerical approximation schemes such as Runge-Kutta methods, the estimation of the error of the NN-based approximations proves to be difficult. In this work, we propose to use the NN's predictions in a high-order implicit Runge-Kutta (IRK) method. The residuals in the implicit system of equations can be related to the NN's prediction error, hence, we can provide an error estimate at several points along a trajectory. We find that this error estimate highly correlates with the NN's prediction error and that increasing the order of the IRK method improves this estimate. We demonstrate this estimation methodology for Physics-Informed Neural Network (PINNs) on the logistic equation as an illustrative example and then apply it to a four-state electric generator model that is regularly used in power system modelling.