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Kernel Methods for Learning About Complex Dynamical Systems


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


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:machine learning; kernel methods; kernel analog forecasting; Gaussian process regression; dynamical systems; SEIR model
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Stuart, Andrew M.
Thesis Committee:
  • Owhadi, Houman (chair)
  • Stuart, Andrew M.
  • Schneider, Tapio
  • Hoffmann, Franca
Defense Date:13 May 2024
Non-Caltech Author Email:dburov (AT)
Record Number:CaltechTHESIS:06082024-115947714
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 5. adapted for Chapter 6.
Burov, Dmitry Anatolyevich0000-0002-5060-6794
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16512
Deposited By: Dmitry Burov
Deposited On:14 Jun 2024 19:18
Last Modified:20 Jun 2024 19:46

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