Jack Xin: Stochastic Interacting Particle Methods and Generative Learning for High Dimensional PDEs

Abstract

Multiscale and high dimensional time dependent partial differential equations (PDE) are challenging to compute by mesh based methods especially when their solutions develop large gradients or concentrations at unknown locations. We disucss stochastic interacting particle (SIP) methods for advection-diffusion-reaction PDEs based on probabilitic representations of solutions, and show their self-adaptivity and efficiency in three and higher dimensions. Using SIP solutions as training data, we compare generative models (such as optimal transport, diffusion, flow-matching and one-step diffusion) in learning, interpolating and predicting solutions as physical parameters vary.