A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators

TL;DR

This paper introduces a new class of logistic functions for better approximation of Koopman operators in nonlinear systems, providing error bounds and tuning parameters, demonstrated on classical models like Van Der Pol oscillator.

Contribution

It proposes a novel class of observable functions that improve finite-dimensional Koopman operator approximations with theoretical error bounds and tunable parameters.

Findings

Enhanced approximation fidelity for Koopman operators.

Derived explicit error bounds for the proposed class.

Validated approach on Van Der Pol oscillator and bistable toggle switch.

Abstract

An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning methods have enabled data-driven discovery of Koopman operators, for both continuous and discrete-time systems. Since Koopman operators are often infinite-dimensional, they are approximated in practice using finite-dimensional systems. The fidelity and convergence of a given finite-dimensional Koopman approximation is a subject of ongoing research. In this paper we introduce a class of Koopman observable functions that confer an approximate closure property on their corresponding finite-dimensional approximations of the Koopman operator. We derive error bounds for the fidelity of this class of observable functions, as well as identify two key learning parameters which can be used to tune performance. We illustrate our approach on two classical nonlinear system models: the Van Der Pol oscillator and the bistable toggle switch.