Kernel Methods for Learning About Complex Dynamical Systems

Citation

Burov, Dmitry Anatolyevich (2024) Kernel Methods for Learning About Complex Dynamical Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/zmmv-1a93. https://resolver.caltech.edu/CaltechTHESIS:06082024-115947714

Abstract

The ubiquitous spread of machine learning tools in natural sciences in recent years has seen trully exponential growth. What sounded like an expression from a sci-fi novel mere 7 years ago, "solving PDEs with machine learning" is hardly surprising to anyone today. The variety of methods is very large, but most of them revolve around the artificial neural networks. Despite tremendous success of applications to problems in natural sciences, and despite many strides towards a fundamental theory of neural networks, they still often lack interpretability and robustness of the results. An alternative, much narrower class of machine learning algorithms is comprised of the kernel methods. These methods, in contrast, offer deep analytical theory, with many approximation results and interpretable components. The firm foundation of the kernel methods, however, is offset by the practical difficulties, such as high computational cost, the burden of high-dimensional optimization and the necessity to manually choose kernel parametrization. This thesis explores a few applications of the kernel methods to dynamical systems, with the goal to address some of those issues. The comparison between the kernel analog forecasting and the plain Gaussian process regression is made, both from theoretical and practical sides, and a parametric extension of the former is proposed. An application of kernel methods to closures of dynamical systems is showcased. Finally, an application of data assimilation machinery to an epidemiological model is shown.

Thesis Files