# Learning Runge-Kutta Integration Schemes for ODE Simulation and Identification

arXiv: https://arxiv.org/abs/2105.04999 · HTML: https://mlsanity.com/paper/2105.04999/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 47 references that are in the map.
- First posted on arXiv 11 May 2021.

## Builds on (most cited first, 8 of 8)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Large-scale Neural Solvers for Partial Differential Equations](https://mlsanity.com/paper/2009.03730/) (2020)
- [ANODEV2: A Coupled Neural ODE Evolution Framework](https://mlsanity.com/paper/1906.04596/) (2019)
- [Bilinear residual Neural Network for the identification and forecasting of dynamical systems](https://mlsanity.com/paper/1712.07003/) (2017)

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

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)

## Abstract (arXiv)

Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be distinct from a true analytical solution. An efficient integration scheme shall further not only provide a trajectory throughout a given state, but also be derived to ensure the generated simulation to be close to the analytical one. Consequently, several integration schemes were developed for different classes of differential equations. Unfortunately, when considering the integration of complex non-linear systems, as well as the identification of non-linear equations from data, this choice of the integration scheme is often far from being trivial. In this paper, we propose a novel framework to learn integration schemes that minimize an integration-related cost function. We demonstrate the relevance of the proposed learning-based approach for non-linear equations and include a quantitative analysis w.r.t. classical state-of-the-art integration techniques, especially where the latter may not apply.