Learning Invariant Subspaces of Koopman Operators--Part 1: A Methodology for Demonstrating a Dictionary's Approximate Subspace Invariance

TL;DR

This paper analyzes how deep learning-based methods approximate Koopman operators by learning dictionaries that exhibit subspace invariance, providing a theoretical basis for their effectiveness in modeling nonlinear dynamics.

Contribution

It introduces the concept of uniform finite approximate closure for dictionary functions and offers a theoretical explanation for deep neural networks' success in Koopman operator approximation.

Findings

DeepDMD learns dictionaries with strong subspace approximation properties.

Error analysis supports the hypothesis of subspace invariance in learned dictionaries.

Theoretical insights explain neural networks' effectiveness in Koopman operator approximation.

Abstract

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, Extended Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators. Part 2 of this paper will extend this explanation by demonstrating the subspace invariant of heterogeneous dictionaries and presenting a head-to-head numerical comparison of deepDMD and low-parameter heterogeneous dictionary learning.