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

Boumediene Hamzi, Houman Owhadi

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.

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

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.

Paper Structure

This paper contains 33 sections, 7 theorems, 52 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. GP Framework Construction
  4. Relationship to Matérn kernels.
  5. Numerical implementation.
  6. Velocity recovery.
  7. Physical interpretation of $\alpha$.
  8. Comparison with standard GP kernels.
  9. The Radon--Nikodym Distortion and Its Implications
  10. Geometric Optimality of the OU Covariance
  11. Predicted Signature: The Optimal $\alpha$ Shift
  12. Future Direction: Learning the Radon--Nikodym Correction
  13. Theoretical Guarantees
  14. Support Alignment and Null Set Preservation
  15. Posterior Consistency
  16. ...and 18 more sections

Key Result

Theorem 2.1

For $\gamma = 1$ and any $\alpha > 0$, equation (2.1) admits a unique invariant measure $\rho$ on $C^{\alpha-\kappa}$ (for any $\kappa > 0$), and $\rho$ is equivalent to $\mu_\alpha$.

Figures (6)

  • Figure 1: Spectral validation for $\alpha \in {1.5, 2.0, 2.5}$. Measured exponents: $-4.14 \pm 0.12$ (theory: $-4.00$), $-5.05 \pm 0.12$ (theory: $-5.00$), $-6.17 \pm 0.12$ (theory: $-6.00$). Differences between $\alpha$ values match theory: $\Delta\beta \approx 1.0$ per unit increase.
  • Figure 2: Statistical robustness of CHT advantage. Mean improvement: $+14.99 \% \pm 16.70 \%$ across 20 field realizations. CHT outperforms RBF in majority of cases, with improvements reaching up to $+30 \%$ in favorable conditions.
  • Figure 3: CHT advantage scales with observation density. Improvements grow from $+9.3\%$ at $m=20$ to $+29.7\%$ at $m=150$, demonstrating that the physics-informed prior becomes more valuable with more constraining data.
  • Figure 4: Optimal $\alpha$ differs from theoretical value. Best performance at $\alpha = 1.25$ ($+15.24\%$ improvement), not the field generation value $\alpha = 1.5$. The framework is robust across $\alpha \in [0.75, 1.5]$, all providing $>+13\%$ improvements.
  • Figure 5: Comprehensive comparison of CHT-GP and RBF-GP performance. (a) Error distribution across all test cases, (b) Scatter plot comparing individual case performance, (c) Error progression across test cases, (d) Example true vorticity field, (e) CHT-GP reconstruction, (f) RBF-GP reconstruction. The CHT-GP framework demonstrates consistent advantages in reconstruction accuracy and reliability.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 2.1: CHT25
  • Theorem 2.2: CHT25
  • Proposition 5.1: Null Set Equivalence
  • proof
  • Corollary 5.2: Support Equality
  • proof
  • Corollary 5.3: Preservation of Typical Events
  • proof
  • Remark 1
  • Theorem 5.4: Posterior Consistency
  • ...and 5 more