Regression-based projection for learning Mori-Zwanzig operators

TL;DR

This paper introduces a regression-based framework for learning Mori-Zwanzig operators, enabling data-driven modeling of dynamical systems with varying complexity and improved accuracy through nonlinear regression models.

Contribution

It presents a unified method to extract Markov and memory operators using regression, bridging the gap between simple and complex projection operators in data-driven dynamical systems modeling.

Findings

Linear regression recovers Mori's projection operator.

Nonlinear models improve approximation accuracy.

Progressive enhancement with more expressive regressions.

Abstract

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural-network-based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.