Core Concepts
The authors show that the Bayesian posterior distribution for the initial condition of the 2D Navier-Stokes equations concentrates around the true initial condition when sufficiently many noisy measurements of the velocity field are available, providing statistical consistency guarantees for Bayesian data assimilation methods in this nonlinear PDE setting.
Abstract
The authors consider a Bayesian data assimilation model for the 2D periodic Navier-Stokes equations, where the initial condition is modeled as a Gaussian process prior. They prove that if the system is updated with enough discrete noisy measurements of the velocity field, the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation.
Key highlights:
The authors provide an explicit quantitative estimate for backward uniqueness of solutions of the 2D Navier-Stokes equations, which is a crucial ingredient in the statistical analysis.
They show that the convergence rate of the posterior to the true initial condition is at best logarithmic in the sample size, but describe conditions on the initial conditions where faster rates are possible.
The authors also establish a minimax lower bound, showing that the logarithmic rates obtained are essentially optimal in an information-theoretic sense.
The results validate the statistical consistency of Bayesian data assimilation algorithms for the 2D Navier-Stokes equations, an important nonlinear PDE model in fluid mechanics and geophysical sciences.