Deterministic particle flows for constraining SDEs

Devising optimal interventions for diffusive systems often requires the solution of the Hamilton-Jacobi-Bellman ({HJB}) equation, a nonlinear backward partial differential equation ({PDE}), that is, in general, nontrivial to solve. Existing control methods either tackle the HJB directly with grid-based PDE solvers, or resort to iterative stochastic path sampling to obtain the necessary controls. Here, we present a framework that interpolates between these two approaches. By reformulating the optimal interventions in terms of logarithmic gradients (\emph{scores}) of two forward probability flows, and by employing deterministic particle methods for solving Fokker-Planck equations, we introduce a novel \emph{deterministic} particle framework that computes the required optimal interventions in \emph{one-shot}.