Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression
2026-06-30 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address the problem of accurately predicting complex systems when the data is noisy. They explain that traditional methods like kernel ridge regression work well on clean data but struggle with noise. To improve this, they use a 'weak formulation,' which helps filter out noise during learning. Combining this with kernel methods, they create a new approach called Weak-form Kernel Ridge Regression (WKRR), which performs better than existing methods on both simple and very complex datasets.
kernel ridge regressionweak formulationnoise filteringdynamical systemsbias-variance decompositionchaotic systemsmachine learningscientific computing
Authors
Max Kreider, John Harlim, Daning Huang
Abstract
Accurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing. Kernel ridge regression learning strategies are often effective when applied to clean data, but have limited success with noisy data. Recent work has observed that a weak formulation can act to filter noisy data, and different learning strategies have achieved increased noise robustness with a weak-form framework. In this manuscript, we give an overview of the filtering mechanism behind the weak formulation and provide a bias-variance error decomposition. Using these insights, we combine a weak formulation with a kernel learning strategy to propose Weak-form Kernel Ridge Regression (WKRR) for learning dynamical systems. The proposed framework is simple to implement, effective for both clean and noisy data, and outperforms several baseline methods. We demonstrate the performance of WKRR on chaotic benchmark systems in up to 64 dimensions, as well as 15,000-dimensional real-world fluid data.