Bayesian Methods for the Navier-Stokes Equations
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| Format: | Preprint |
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2026
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| author | Polson, Nicholas Sokolov, Vadim |
| author_facet | Polson, Nicholas Sokolov, Vadim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_02945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bayesian Methods for the Navier-Stokes Equations Polson, Nicholas Sokolov, Vadim Computation Numerical Analysis 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. |
| title | Bayesian Methods for the Navier-Stokes Equations |
| topic | Computation Numerical Analysis |
| url | https://arxiv.org/abs/2602.02945 |