  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.
