Preserving Nonlinear Constraints in Variational Flow Filtering Data Assimilation

TL;DR

This paper extends variational Fokker-Planck particle flow filters to incorporate nonlinear physical constraints, ensuring that the estimated states in data assimilation remain physically feasible within the constrained manifold.

Contribution

It introduces two novel algorithms, VFPSTAB and VFPDAE, for preserving nonlinear constraints in particle flow data assimilation, with the latter exactly maintaining constraints via stochastic differential-algebraic equations.

Findings

VFPDAE exactly preserves nonlinear constraints in test problems.

The methods improve physical consistency of state estimates.

Demonstrated on systems like double pendulum and Navier-Stokes equations.

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.