Deterministic particle flows for constraining SDEs

TL;DR

This paper introduces a deterministic particle flow framework for solving stochastic differential equations (SDEs) to compute optimal interventions, bridging the gap between PDE solvers and stochastic sampling methods.

Contribution

It presents a novel deterministic approach that uses particle methods and score functions to efficiently solve control problems for diffusive systems.

Findings

Provides a one-shot computation of optimal interventions.

Bridges the gap between grid-based PDE solvers and stochastic sampling methods.

Offers a fully deterministic framework for SDE control.

Abstract

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 ( scores ) of two forward probability flows, and by employing deterministic particle methods for solving Fokker-Planck equations, we introduce a novel fully deterministic framework that computes the required optimal interventions in one shot.