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：我们研究了周期边界条件下二维纳维-Stokes方程的非线性贝叶斯数据同化模型，其中初始条件采用高斯过程先验建模。我们证明，若系统通过足够多的离散含噪速度场观测值进行更新，则后验分布最终将集中于时间演化方程真实解附近，且后验平均向量场能够一致地还原初始条件。进一步研究表明，收敛速度在一般情况下不可能快于样本量的对数倒数阶，但给出了初始条件满足特定要求时可获得更快收敛速度的条件。在证明过程中，我们为二维纳维-斯托克斯方程解的后向唯一性提供了显式定量估计。