A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity

TL;DR

This paper introduces a Gaussian process framework for 2D stochastic Navier-Stokes equations, grounded in recent theoretical proof of quasi-Gaussianity, providing a rigorous long-term dynamical prior for turbulent flow modeling.

Contribution

It constructs a Gaussian process prior based on the stationary covariance of the linear Ornstein-Uhlenbeck model, linking SPDE theory with practical data assimilation for turbulent flows.

Findings

Provides a principled GP prior for 2D SNS based on theoretical foundations

Bridges SPDE theory and data assimilation in fluid dynamics

Offers a rigorous long-term dynamical justification for turbulence modeling

Abstract

The recent proof of quasi-Gaussianity for the 2D stochastic Navier--Stokes (SNS) equations by Coe, Hairer, and Tolomeo establishes that the system's unique invariant measure is equivalent (mutually absolutely continuous) to the Gaussian measure of its corresponding linear Ornstein--Uhlenbeck (OU) process. While Gaussian process (GP) frameworks are increasingly used for fluid dynamics, their priors are often chosen for convenience rather than being rigorously justified by the system's long-term dynamics. In this work, we bridge this gap by introducing a probabilistic framework for 2D SNS built directly upon this theoretical foundation. We construct our GP prior precisely from the stationary covariance of the linear OU model, which is explicitly defined by the forcing spectrum and dissipation. This provides a principled, GP prior with rigorous long-time dynamical justification for turbulent flows, bridging SPDE theory and practical data assimilation.