Core Concepts
Identifying reduced order models using invariant foliations in forced systems.
Abstract
The content discusses the process of identifying reduced order models (ROM) in forced systems using invariant foliations. It outlines a four-step process involving identifying an approximate invariant torus, defining linear dynamics, establishing globally defined invariant foliations, and extracting the invariant manifold. The limitations of fitting invariant manifolds to data are highlighted, requiring further mathematical resolution. Various architectural approaches like autoencoders and equation-free models are compared with invariant foliations. The importance of identifying genuine ROMs that discard unimportant dynamics is emphasized. The paper also delves into numerical methods for ROM identification and illustrates the method on examples.
Stats
"A ROM represents dynamics independent of the coordinate system."
"An integration constant c is used in trajectory calculations."
"The error near equilibrium due to lack of required differentiability."
"Data fitting approximates the invariance equation at each data point."
"Global criterion for defining unique and meaningful invariant manifolds."
Quotes
"We identify reduced order models (ROM) of forced systems from data using invariant foliations."
"Autoencoders cannot ensure invariance except when system dimensionality matches ROM dimensionality."
"Invariant foliations are used to analyze chaotic systems and find initial conditions for ROMs."