# Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

arXiv: https://arxiv.org/abs/2110.00546 · HTML: https://mlsanity.com/paper/2110.00546/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 81 references that are in the map.
- First posted on arXiv 1 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Using machine learning to correct model error in data assimilation and forecast applications](https://mlsanity.com/paper/2010.12605/) (2020)
- [Combining data assimilation and machine learning to infer unresolved scale parametrisation](https://mlsanity.com/paper/2009.04318/) (2020)
- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Deep Neural Networks for Data-Driven Turbulence Models](https://mlsanity.com/paper/1806.04482/) (2018)
- [Analog forecasting of extreme-causing weather patterns using deep learning](https://mlsanity.com/paper/1907.11617/) (2019)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Data Assimilation: A Mathematical Introduction](https://mlsanity.com/paper/1506.07825/) (2015)
- [Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations](https://mlsanity.com/paper/2010.00072/) (2020)
- [Model Reduction with Memory and the Machine Learning of Dynamical Systems](https://mlsanity.com/paper/1808.04258/) (2018)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Climbing down Charney's ladder: Machine Learning and the post-Dennard era of computational climate science](https://mlsanity.com/paper/2005.11862/) (2020)
- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Data-driven super-parameterization using deep learning: Experimentation with multi-scale Lorenz 96 systems and transfer-learning](https://mlsanity.com/paper/2002.11167/) (2020)
- [Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction](https://mlsanity.com/paper/1904.10904/) (2019)
- [Sparse Methods for Automatic Relevance Determination](https://mlsanity.com/paper/2005.08741/) (2020)

## Built on by (most cited first, 6 of 6)

- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems](https://mlsanity.com/paper/2206.04811/) (2022)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence](https://mlsanity.com/paper/2509.20365/) (2025)
- [Where Is My Physics Wrong? Localized and Identifiable Discovery of Model Discrepancy](https://mlsanity.com/paper/2606.23215/) (2026)

## Abstract (arXiv)

Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation (DA) technique such as ensemble Kalman filter (EnKF) is employed to provide the analysis state of the system, which is then used to estimate the model error. Finally, an equation-discovery technique, here the relevance vector machine (RVM), a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, and closed-form representation of the model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.