Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds

Andreas Bittracher*, Stefan Klus, Boumediene Hamzi, Péter Koltai, Christof Schütte

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
23 Downloads (Pure)

Abstract

We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to the previous parameterization approaches.

Original languageEnglish
Article number3
JournalJournal of Nonlinear Science
Volume31
Issue number1
Early online date18 Dec 2020
DOIs
Publication statusPublished - Feb 2021

ASJC Scopus subject areas

  • Modelling and Simulation
  • General Engineering
  • Applied Mathematics

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