Path Integral Diffusion: From Integrable Bridge to Adaptive, Guided, and Mean-Field Sampling

Date

Jun 26, 2026

Time

2:00 PM - 3:00 PM

Speaker

Prof. Michael Chertkov

Affiliation

University of Arizona

Language

en

Main Topic

Physik

Other Topics

Physik

Description

Diffusion models of Generative AI are usually introduced through learned score fields and reverse-time stochastic dynamics. In this talk, I will present an alternative, control-theoretic viewpoint in which generative sampling is formulated as a bridge-diffusion problem and solved through Path Integral Diffusion (PID). In this framework, sampling is recast as stochastic optimal transport with potential, and the optimal drift admits an explicit representation through forward and backward Green functions. For harmonic potentials, this yields an analytic and interpretable construction in which the score, drift, and predicted terminal state can all be written in closed form. I will then discuss two natural extensions. First, in Adaptive PID, the quadratic stiffness of the potential is allowed to vary in time. With piecewise-constant schedules and Gaussian-mixture targets, the resulting bridge remains analytically tractable, making it possible to optimize transient behavior rather than only terminal accuracy. This leads to a new perspective on quality of sampling in terms of pathwise diagnostics such as sensitivity of the drift. Second, in Guided PID, the center of the quadratic potential is also allowed to move, providing a mechanism to steer flows in probability space and, in some applications, in physical space as well. These constructions lead naturally to Mean-Field PID, where samples no longer evolve independently but interact through a self-consistent mean field. This produces a cooperative version of bridge diffusion, connecting generative modeling with McKean–Vlasov dynamics and mean-field control. The overall message is that one can move from standard diffusion sampling to adaptive, guided, and ultimately cooperative sampling while preserving a large degree of analytic control and interpretability. Dr. Michael "Misha" Chertkov is a Professor of Mathematics and Chair of the Graduate Interdisciplinary Program in Applied Mathematics at the University of Arizona. His research addresses foundational challenges in mathematics, statistics, machine learning, and artificial intelligence, particularly as they apply to and are inspired by physical systems like fluid mechanics. He also works on applications in the control of engineered systems, such as energy grids, and bio-social systems. Dr. Chertkov received his Ph.D. in physics from the Weizmann Institute of Science in 1996. After obtaining his Ph.D., he spent three years as a R.H. Dicke Fellow in the Department of Physics at Princeton University. In 1999, he joined the Los Alamos National Laboratory, first as a J.R. Oppenheimer Fellow, later becoming a Technical Staff Member in the Theory Division. He transitioned to the University of Arizona in 2019. Throughout his career, Dr. Chertkov has contributed about 300 research papers. He holds the title of Fellow in both the AAAS and the American Physical Society and is a Senior Member of IEEE.

Last modified: Jun 22, 2026, 7:40:21 AM