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

arXiv: https://arxiv.org/abs/2511.21281 · HTML: https://mlsanity.com/paper/2511.21281/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 26 Nov 2025.

## Builds on (most cited first, 7 of 7)

- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Gaussian Process Hydrodynamics](https://mlsanity.com/paper/2209.10707/) (2022)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields](https://mlsanity.com/paper/2507.17582/) (2025)

## Abstract (arXiv)

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.