Deterministic particle flows for constraining SDEs

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.

4+3 pages, 2 figures -- workshop paper -- Machine Learning and the Physical Sciences, Workshop at the 35th Conference on Neural Information Processing Systems (NeurIPS 2021) The related journal paper is published here: https://link.aps.org/doi/10.1103/PhysRevResearch.4.043035

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