TY - JOUR
T1 - Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds
AU - Bittracher, Andreas
AU - Klus, Stefan
AU - Hamzi, Boumediene
AU - Koltai, Péter
AU - Schütte, Christof
N1 - Funding Information:
The authors would like to thank the anonymous reviewers for constructive comments and suggestions that helped to improve the paper. This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 “Scaling Cascades in Complex Systems”, Project Number 235221301, Projects A01, B03, and B06.
Publisher Copyright:
© 2020, The Author(s).
PY - 2021/2
Y1 - 2021/2
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85097774827&partnerID=8YFLogxK
U2 - 10.1007/s00332-020-09668-z
DO - 10.1007/s00332-020-09668-z
M3 - Article
AN - SCOPUS:85097774827
SN - 0938-8974
VL - 31
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 1
M1 - 3
ER -