# Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction

arXiv: https://arxiv.org/abs/1904.10904 · HTML: https://mlsanity.com/paper/1904.10904/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2019.

## Built on by (most cited first, 15 of 15)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Bridging observation, theory and numerical simulation of the ocean using Machine Learning](https://mlsanity.com/paper/2104.12506/) (2021)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [Machine Learning Emulation of 3D Cloud Radiative Effects](https://mlsanity.com/paper/2103.11919/) (2021)
- [Benchmark Dataset for Precipitation Forecasting by Post-Processing the Numerical Weather Prediction](https://mlsanity.com/paper/2206.15241/) (2022)
- [Interpretable Machine Learning for Weather and Climate Prediction: A Survey](https://mlsanity.com/paper/2403.18864/) (2024)
- [Programming with Neural Surrogates of Programs](https://mlsanity.com/paper/2112.06148/) (2021)
- [History-Based, Bayesian, Closure for Stochastic Parameterization: Application to Lorenz '96](https://mlsanity.com/paper/2210.14488/) (2022)
- [Development of an offline and online hybrid model for the Integrated Forecasting System](https://mlsanity.com/paper/2403.03702/) (2024)
- [Variability of echo state network prediction horizon for partially observed dynamical systems](https://mlsanity.com/paper/2306.10797/) (2023)
- [Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systems](https://mlsanity.com/paper/2512.11834/) (2025)

## Abstract (arXiv)

Dynamical weather and climate prediction models underpin many studies of the Earth system and hold the promise of being able to make robust projections of future climate change based on physical laws. However, simulations from these models still show many differences compared with observations. Machine learning has been applied to solve certain prediction problems with great success, and recently it's been proposed that this could replace the role of physically-derived dynamical weather and climate models to give better quality simulations. Here, instead, a framework using machine learning together with physically-derived models is tested, in which it is learnt how to correct the errors of the latter from timestep to timestep. This maintains the physical understanding built into the models, whilst allowing performance improvements, and also requires much simpler algorithms and less training data. This is tested in the context of simulating the chaotic Lorenz '96 system, and it is shown that the approach yields models that are stable and that give both improved skill in initialised predictions and better long-term climate statistics. Improvements in long-term statistics are smaller than for single time-step tendencies, however, indicating that it would be valuable to develop methods that target improvements on longer time scales. Future strategies for the development of this approach and possible applications to making progress on important scientific problems are discussed.