Data-driven model reduction for stochastic Burgers equations

TL;DR

This paper develops efficient data-driven closure models for 1D stochastic Burgers equations using statistical learning, enabling significant space-time reduction while accurately capturing key statistical properties.

Contribution

It introduces nonlinear autoregression models trained from data to effectively reduce the stochastic Burgers equations in space and time.

Findings

NAR models accurately reproduce energy spectra and invariant densities.

Space reduction is unlimited with NAR models using two Fourier modes.

Time reduction is limited by stability, with a criterion for optimal reduction.

Abstract

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variables' trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal and optimal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model's stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step where the K-mode Galerkin system's mean CFL number agrees with the full model's.