Data-driven non-Markovian closure models

TL;DR

This paper introduces multilayer stochastic models (MSMs) for stable, data-driven closure modeling of high-dimensional systems, compares them with Mori-Zwanzig formalism predictions, and demonstrates their effectiveness on climate and population models.

Contribution

It develops a new multilayer stochastic modeling framework that generalizes existing approaches and connects with the Mori-Zwanzig formalism for non-Markovian systems.

Findings

MSMs can approximate the generalized Langevin equation effectively.

A correlation-based criterion assesses MSM approximation quality.

MSMs capture key statistical features in climate and population models.

Abstract

This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori-Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR-MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a very broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The approach is applied to conceptual nonlinear models borrowed from climate dynamics and population dynamics. In both cases, it is shown that the resulting closure models are able to capture the main statistical features of the dynamics, even in presence of weak time-scale separation.