Details

Autor(en) / Beteiligte

• Meuris, Brek

Titel

Model Reduction for Dynamical Systems: Machine Learning and Memory

Ort / Verlag

ProQuest Dissertations & Theses

Erscheinungsjahr

2022

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• Many of the physical phenomena arising in science and engineering can be described by partial differential equations. Typically, partial differential equations of interest cannot be solved analytically and require the use of scientific computing methods, machine learning methods, or a combination of these methods to approximate the solution. Spectral methods are one technique commonly utilized by the scientific computing community to solve partial differential equations due to their ability to achieve a high order of convergence; however, spectral methods require an appropriate choice of basis function--which is not always obvious--and result in a high-dimensional system of equations that must be solved. Reduced order modeling seeks to address the high-dimensionality of these systems of equations and reproduce the dominant features of the solution while utilizing a lower-dimensional system. For systems that are multiscale in nature, reduced order modeling becomes more nuanced, as it is no longer feasible to simply truncate and evolve just a subset of the degrees of freedom. The focus of this work is the development of two frameworks for the evolution of partial differential equations. The first is a reduced order modeling framework for multiscale dynamical systems that is based on the Mori-Zwanzig formalism. As part of this work, a parameter is introduced to control the time decay of the memory term, which results from the application of the Mori-Zwanzig formalism, and is selected based on data obtained from a well-resolved full simulation. This framework is applied first to the singular, one-dimensional inviscid Burgers equation as a proof-of-concept and then extended to the three-dimensional Euler equations, where full simulations are only feasible for short times. The second is a scientific machine learning framework that utilizes deep operator neural networks (DeepONets) for the identification of candidate basis functions. The candidate basis functions--which are custom-made for the dynamical system of interest--are orthonormalized through the singular value decomposition and are suitable for usage in a spectral method. Model reduction is possible in this framework through the specification of a singular value threshold, thereby allowing a lower-dimensional system to be utilized for the evolution of a partial differential equation. This framework is applied to six one-dimensional models to provide a proof-of-concept: the advection, advection-diffusion, viscous Burgers, Korteweg-de Vries, Kuramoto-Sivashinsky, and inviscid Burgers equations.