Repository: Freie Universit├Ąt Berlin, Math Department

Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

Koronaki, D. Eleni and Evangelou, Nikolaos and Martin-Linares, Cristina P. and Titi, Edriss and Kevrekidis, Ioannis G. (2023) Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era. Preprint . (Unpublished)

[img] PDF
7MB

Official URL: https://doi.org/10.48550/arXiv.2310.15816

Abstract

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.

Item Type:Article
Subjects:Mathematical and Computer Sciences
Mathematical and Computer Sciences > Mathematics
Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:3034
Deposited By: Jana Jerosch
Deposited On:17 Jan 2024 08:47
Last Modified:17 Jan 2024 08:47

Repository Staff Only: item control page