Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case

TL;DR

This paper introduces a novel method for deriving non-Markovian reduced systems for SPDEs with additive noise using stochastic parameterizing manifolds, improving modeling accuracy by capturing memory effects.

Contribution

The authors develop a new approach to approximate small-scale dynamics in SPDEs via stochastic PMs, enabling efficient non-Markovian reduced models that outperform traditional methods.

Findings

Effective derivation of non-Markovian reduced systems for SPDEs

Demonstrated improved modeling performance on stochastic Burgers-type equation

Showed the importance of non-Markovian features in reduced models

Abstract

This article proposes for stochastic partial differential equations (SPDEs) driven by additive noise, a novel approach for the approximate parameterizations of the ``small'' scales by the ``large'' ones, along with the derivaton of the corresponding reduced systems. This is accomplished by seeking for stochastic parameterizing manifolds (PMs) introduced in a previous work by the authors(*), which are random manifolds aiming to provide --- in a mean square sense --- such approximate parameterizations. Backward-forward systems are designed to give access to such PMs as pullback limits depending through the nonlinear terms on the time-history of the dynamics of the low modes when the latter is simply approximated by its stochastic linear component. It is shown that the corresponding pullback limits can be efficiently determined, leading in turn to an operational procedure for the derivation of non-Markovian reduced systems able to achieve good modeling performances in practice. This is illustrated on a stochastic Burgers-type equation, where it is shown that the corresponding non-Markovian features of these reduced systems play a key role to reach such performances.