# Neural Network Representation of Time Integrators

arXiv: https://arxiv.org/abs/2211.17039 · HTML: https://mlsanity.com/paper/2211.17039/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 18 references that are in the map.
- First posted on arXiv 30 Nov 2022.

## Builds on (most cited first, 6 of 6)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Fractional Deep Neural Network via Constrained Optimization](https://mlsanity.com/paper/2004.00719/) (2020)
- [Novel Deep neural networks for solving Bayesian statistical inverse](https://mlsanity.com/paper/2102.03974/) (2021)
- [NINNs: Nudging Induced Neural Networks](https://mlsanity.com/paper/2203.07947/) (2022)

## Built on by (most cited first, 1 of 1)

- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)

## Abstract (arXiv)

Deep neural network (DNN) architectures are constructed that are the exact equivalent of explicit Runge-Kutta schemes for numerical time integration. The network weights and biases are given, i.e., no training is needed. In this way, the only task left for physics-based integrators is the DNN approximation of the right-hand side. This allows to clearly delineate the approximation estimates for right-hand side errors and time integration errors. The architecture required for the integration of a simple mass-damper-stiffness case is included as an example.