We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier-Stokes equations.

翻译：我们考虑一个非线性贝叶斯数据同化模型，该模型作用于带有高斯过程先验初始条件的周期性二维Navier-Stokes方程。研究表明，如果系统通过足够多的速度场离散噪声观测进行更新，那么后验分布最终将集中于时间演化方程真实解附近，特别地，后验均值向量场能够一致地恢复初始条件。我们进一步证明，收敛速率通常不会快于样本量的对数倒数，但描述了在特定初始条件下可实现更快收敛速率的情形。在证明过程中，我们给出了二维Navier-Stokes方程向后唯一性的显式定量估计。