# Learning Closure of Dynamical Systems with Kernel Ridge Regression

arXiv: https://arxiv.org/abs/2610.02564 · HTML: https://mlsanity.com/paper/2610.02564/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 34 references that are in the map.
- First posted on arXiv 1 Oct 2026.

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

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure](https://mlsanity.com/paper/2105.05690/) (2021)
- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [Hyperbolic Machine Learning Moment Closures for the BGK Equations](https://mlsanity.com/paper/2401.04783/) (2024)
- [Learning solution operator of dynamical systems with diffusion maps kernel ridge regression](https://mlsanity.com/paper/2512.17203/) (2025)
- [The Machine Learning Approach to Moment Closure Relations for Plasma: A Review](https://mlsanity.com/paper/2511.22486/) (2025)
- [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026)

## Abstract (arXiv)

We develop a closure modeling framework for identifying missing components of dynamical systems using Kernel Ridge Regression (KRR). The framework addresses two classes of closure problems: difference-equation closures arising in ODE and PDE settings, and algebraic closures arising from moment closure in kinetic equations. For the first class, we derive an error bound in an ODE setting that quantifies contributions from time integration, approximation of unresolved scales, and interpolation required to couple unresolved-scale effects to the resolved solver. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate long-horizon predictions and substantial improvements over an LSTM-based closure model. For the second class, we consider moment closure for a one-dimensional kinetic equation by modeling discrepancies between kinetic and macroscopic fluxes as a function of the resolved macroscopic variables. We compare global KRR models based on PCA coordinates with spatially local models. While the global model performs well for unimodal initial conditions, its accuracy deteriorates for bimodal initial conditions. Spatially local models with appropriate modeling inputs improve robustness and achieve higher predictive accuracy.