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On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations

Richard Nickl, Edriss S. Titi

TL;DR

The paper addresses posterior consistency for data assimilation in the periodic $2D$ Navier–Stokes equations when the initial condition is given a Gaussian process prior. It develops forward-backward stability results for the Navier–Stokes forward map and leverages Bayesian non-linear inversion theory to prove posterior contraction around the true initial state and its induced flow under increasing, discrete, noisy velocity measurements; the typical rate is logarithmic in the sample size with potential improvements under spectral (Stokes) constraints. Key contributions include an explicit quantitative backward-uniqueness stability estimate, a sharpness result showing the logarithmic rate limit, and minimax lower bounds that match the upper-rate under general conditions. The findings provide principled, prior-robust guarantees for nonlinear PDE data assimilation and clarify information-theoretic limits for recovering initial data from partial observations in dissipative fluid dynamics.

Abstract

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.

On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations

TL;DR

The paper addresses posterior consistency for data assimilation in the periodic Navier–Stokes equations when the initial condition is given a Gaussian process prior. It develops forward-backward stability results for the Navier–Stokes forward map and leverages Bayesian non-linear inversion theory to prove posterior contraction around the true initial state and its induced flow under increasing, discrete, noisy velocity measurements; the typical rate is logarithmic in the sample size with potential improvements under spectral (Stokes) constraints. Key contributions include an explicit quantitative backward-uniqueness stability estimate, a sharpness result showing the logarithmic rate limit, and minimax lower bounds that match the upper-rate under general conditions. The findings provide principled, prior-robust guarantees for nonlinear PDE data assimilation and clarify information-theoretic limits for recovering initial data from partial observations in dissipative fluid dynamics.

Abstract

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.
Paper Structure (10 sections, 8 theorems, 92 equations)

This paper contains 10 sections, 8 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Forward and inverse stability in the $2D$ Navier-Stokes equations
  4. Non-linear data assimilation with Gaussian process priors
  5. Proofs
  6. Proof of Theorem \ref{['main1']}
  7. Proof of Theorem \ref{['stabsharp']}
  8. Proof of Theorems \ref{['mainstat']} and \ref{['fastrates']}
  9. Proof of Theorem \ref{['minmax']}
  10. Proof of Proposition \ref{['classical']}

Key Result

Proposition 1

Let $T>0$ and let $u(0) \in V$ satisfy $\|u(0)\|_{V} \le U$ for some $U>0$. A) The two-dimensional periodic Navier-Stokes equations (nstokes) have a unique strong solution $u \in C([0,T], V) \cap L^2((0,T], \mathcal{D}(A))$ with $du/dt \in L^2((0,T], H)$. There exists a constant $c_U\equiv c(U, \|f\ Moreover, for every $m>0$ there exists $c=c(m, \|f\|_{H^1}, \nu, T)>0$ such that we have B) If $v(

Theorems & Definitions (12)

  • Proposition 1
  • Theorem 1
  • Remark 1: Inverse Poincaré inequality and the Stokes spectrum
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 2: MAP Estimates
  • Lemma 1
  • proof
  • ...and 2 more