Learning Invariant Subspaces of Koopman Operators--Part 2: Heterogeneous Dictionary Mixing to Approximate Subspace Invariance

TL;DR

This paper introduces a method for mixing heterogeneous dictionaries, combining different nonlinear functions, to efficiently approximate Koopman operators with high accuracy and interpretability, rivaling deep learning approaches.

Contribution

It demonstrates that mixing heterogeneous dictionaries like SILL and RBFs can match deepDMD accuracy with fewer parameters, enhancing interpretability and efficiency.

Findings

Heterogeneous dictionary mixing achieves deepDMD accuracy.

Mixed dictionaries require fewer parameters.

Method maintains geometric interpretability.

Abstract

This work builds on the models and concepts presented in part 1 to learn approximate dictionary representations of Koopman operators from data. Part I of this paper presented a methodology for arguing the subspace invariance of a Koopman dictionary. This methodology was demonstrated on the state-inclusive logistic lifting (SILL) basis. This is an affine basis augmented with conjunctive logistic functions. The SILL dictionary's nonlinear functions are homogeneous, a norm in data-driven dictionary learning of Koopman operators. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.