Optimal Parameterizing Manifolds for Anticipating Tipping Points and Higher-order Critical Transitions

TL;DR

This paper introduces an optimal parameterizing manifold (OPM) approach for low-order system reduction, enabling accurate anticipation of tipping points and critical transitions by optimizing parameterizations based on data and model dynamics.

Contribution

It develops a variational method for deriving OPMs that generalize invariant manifolds, especially near instability onset, to improve prediction of critical transitions.

Findings

OPMs can accurately predict higher-order critical transitions.

Optimizing backward integration time enhances parameterization accuracy.

The method effectively anticipates catastrophic tipping points.

Abstract

A general, variational approach to derive low-order reduced systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes the more classical notions of invariant or slow manifold when breakdown of "slaving" occurs, i.e. when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through integration of auxiliary backward-forward (BF) systems built from the model's equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact, away from instability onset, due to breakdown of slaving typically encountered e.g. for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown, through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions take place.