Preserving Nonlinear Constraints in Variational Flow Filtering Data Assimilation

TL;DR

This work extends the Variational Fokker-Planck (VFP) framework for data assimilation to preserve nonlinear state constraints by introducing two constraint-preserving variants: VFPSTAB, which adds a stabilizing drift for inexact constraint maintenance, and VFPDAE, which enforces constraints exactly through a stochastic differential-algebraic equation (SDAE) formulation. An implicit-explicit time integration scheme is developed for the SDAE, and covariance regularization is used to prevent filter divergence. The methods are tested on three challenging systems—the double pendulum, the Korteweg-de Vries equation, and the incompressible Navier–Stokes equations—showing that VFPDAE achieves exact constraint preservation and strong performance, while VFPSTAB offers a computationally cheaper alternative with controlled inexactness. Overall, the paper demonstrates that incorporating constraint-preserving dynamics into particle flow filters yields physically consistent analyses and can outperform standard filters in preserving nonlinear invariants, with practical implications for geophysical and fluid-dynamics data assimilation.

Abstract

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP) -- a generic particle flow filtering framework -- is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

Figures (5)

• Figure 1: A simple pendulum example where the forecast ensemble members lie on the constraint manifold, but the standard ETKF analysis does not. The goal is to obtain an analysis ensemble that lives on the constraint manifold like the Constrained ETKF.
• Figure 2: Results for the double pendulum experiments.
• Figure 3: Results for the Korteweg-deVries experiments where the VFP methods outperform the ETKF methods.
• Figure 4: Incompressible Navier-Stokes: RMSE for different methods.
• Figure 5: CRMSE results for the Navier-Stokes experiments.

Theorems & Definitions (5)

• Example 1: Pendulum
• Example 2: Gaussian case
• Remark 2
• Remark 3
• Remark 4