HJ-Sampler - A Bayesian Sampler for Inverse Problems of Stochastic Processes by Leveraging Hamilton-Jacobi PDEs

Explore a mathematical conference talk presenting the HJ-sampler, a novel Bayesian sampling algorithm for inverse problems in stochastic processes that leverages Hamilton-Jacobi partial differential equations and score-based generative models. Learn how the research extends the Cole-Hopf transform within an abstract framework incorporating linear operators, revealing connections between optimal transport and Bayesian inference under specific boundary conditions. Discover the two-stage algorithm that first solves viscous Hamilton-Jacobi PDEs and then samples from the associated stochastic optimal control problem, with flexibility in numerical solver selection. Examine two solver variants: the Riccati-HJ-sampler using the Riccati method and the SGM-HJ-sampler utilizing diffusion models. Understand the theoretical foundations linking stochastic processes, optimal control, and diffusion models, particularly how the linear operator can serve as either the infinitesimal generator of a stochastic process or its adjoint for Bayesian inference applications. Review numerical examples demonstrating the effectiveness of these proposed methods for solving inverse problems of stochastic differential equations with given terminal observations, presented as part of collaborative research in scientific machine learning.