Bayesian Methods for the Navier-Stokes Equations

TL;DR

This paper introduces a Bayesian framework for solving the Navier-Stokes equations numerically, providing uncertainty quantification and stable parameter learning through particle methods, applicable in both 2D and 3D cases.

Contribution

It develops a Bayesian approach treating discretized Navier-Stokes as a state-space model, enabling uncertainty quantification and stable parameter learning with particle methods.

Findings

Monte Carlo solvers for 2D Navier-Stokes with uncertainty propagation

Particle learning enables stable parameter estimation

Framework supports sequential observational updates

Abstract

We develop a Bayesian methodology for numerical solution of the incompressible Navier--Stokes equations with quantified uncertainty. The central idea is to treat discretized Navier--Stokes dynamics as a state-space model and to view numerical solution as posterior computation: priors encode physical structure and modeling error, and the solver outputs a distribution over states and quantities of interest rather than a single trajectory. In two dimensions, stochastic representations (Feynman--Kac and stochastic characteristics for linear advection--diffusion with prescribed drift) motivate Monte Carlo solvers and provide intuition for uncertainty propagation. In three dimensions, we formulate stochastic Navier--Stokes models and describe particle-based and ensemble-based Bayesian workflows for uncertainty propagation in spectral discretizations. A key computational advantage is that parameter learning can be performed stably via particle learning: marginalization and resample--propagate (one-step smoothing) constructions avoid the weight-collapse that plagues naive sequential importance sampling on static parameters. When partial observations are available, the same machinery supports sequential observational updating as an additional capability. We also discuss non-Gaussian (heavy-tailed) error models based on normal variance-mean mixtures, which yield conditionally Gaussian updates via latent scale augmentation.