Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

TL;DR

This paper introduces a new consistency index based on forward-backward EDMD that effectively measures the quality of the function space for dynamical system modeling, surpassing residual error in reliability.

Contribution

The paper proposes a novel consistency index derived from forward-backward EDMD, which accurately assesses function space quality and is invariant to basis choice.

Findings

Consistency index correlates with prediction accuracy.

It provides a tight upper bound for prediction error.

It is invariant under basis transformations.

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.