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

2025-11-26

AI Summary

Gaussian process (GP) priors commonly employed in modeling two-dimensional stochastic Navier–Stokes (SNS) equations lack dynamical justification, leading to misaligned statistical behavior and compromised uncertainty quantification. Method: We propose a dynamics-driven GP framework grounded in quasi-Gaussian theory and stochastic partial differential equation (SPDE) invariant measure analysis. Specifically, we construct a GP prior whose stationary covariance exactly matches that of the linear Ornstein–Uhlenbeck process governing the system’s long-term dynamics—thereby embedding physical constraints directly into the prior. This marks the first integration of invariant measure theory into GP prior design, enabling principled derivation of the covariance structure from the underlying dynamical equations. Contribution: We establish the first GP modeling paradigm for 2D SNS systems with rigorous theoretical guarantees on asymptotic statistical behavior. Experiments demonstrate substantial improvements in data assimilation accuracy and reliability of uncertainty quantification compared to conventional GP approaches.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• Develops Gaussian Process prior for 2D stochastic Navier-Stokes equations
• Bridges theoretical quasi-Gaussianity proof with practical data assimilation
• Constructs principled prior using linear Ornstein-Uhlenbeck stationary covariance

Innovation

• GP prior based on linear OU stationary covariance
• Framework justified by long-term dynamical theory
• Bridging SPDE theory with data assimilation