Kernel ridge regression models improve dynamical system predictions
Learning Closure of Dynamical Systems with Kernel Ridge Regression
Machine Learning
Summary
Predicting the future behavior of complex systems can be tricky when some parts of the system are unknown or too small to track. The authors use a mathematical tool called Kernel Ridge Regression to learn and fix these missing parts. They show their method works well on classic systems that scientists study, and it does better than some other machine learning methods. This helps make longer and more accurate predictions for both general and specialized types of systems.
What this means in practice
- •For climate model developers: Use kernel ridge regression to improve missing process approximations in large-scale climate simulations for more accurate long-term forecasts.
- •For computational fluid dynamics engineers: Incorporate local kernel ridge regression closures to better capture unresolved scale effects in fluid flow simulations, reducing errors in predictive models.
Authors
Evan Habbershaw, John Harlim, Senwei Liang
Abstract
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.