Stochastic Interacting Particle Methods and Generative Learning for Multiscale PDEs

-

B1 L3 R3119

This talk discusses stochastic interacting particle (SIP) methods for advection-diffusion-reaction PDEs based on probabilistic representations of solutions, and shows their self-adaptivity and efficiency in several space dimensions.

Multiscale 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 discuss stochastic interacting particle (SIP) methods for advection-diffusion-reaction PDEs based on probabilistic representations of solutions, and show their self-adaptivity and efficiency in several space dimensions. Using SIP solutions as training data, we compare generative AI models (optimal transport, diffusion, flow-matching and one-step diffusion) in learning, interpolating and predicting solutions as physical parameters vary.

Presenters

Jack Xin, Distinguished Professor, Department of Mathematics, School of Physical Sciences, University of California, Irvine

Brief Biography

Jack Xin received his Ph.D. degree in mathematics from the Courant Institute of Mathematical Sciences, New York University, in 1990. He was a faculty member at the University of Arizona from 1991 to 1999, and at the University of Texas at Austin from 1999 to 2005. He is currently a Distinguished Professor of Mathematics at the University of California, Irvine. His research interests include applied analysis, computational methods, and their applications in multiscale problems, data science, and AI. He is a fellow of the Guggenheim Foundation, the American Mathematical Society, the American Association for the Advancement of Science, the Society for Industrial and Applied Mathematics, and the Asia-Pacific Artificial Intelligence Association. He received the Qualcomm Faculty Award from 2019 to 2022 and the Qualcomm Gift Award from 2023 to 2025. In 2025, he was elected a member of the National Academy of Artificial Intelligence and received the Academy's Distinguished AI Scholar Award.