Collaborative Research: Towards High-Fidelity Generative Mean Flow Models: A Theory-Validation Feedback Loop for Scientific Machine Learning

This project advances the mission of the National Science Foundation by bridging artificial intelligence (AI) and computational mathematics to solve critical engineering challenges in fusion energy and advanced materials. Generative AI models have revolutionized computer science tasks such as image generation, but their full potential in scientific and engineering fields remains unlocked due to a lack of accuracy guarantees and high computational costs. This project addresses these challenges by developing a new class of highly accurate and fast AI models designed specifically for physical simulations. By accelerating complex computations, the research aims to drive innovations in fields of federal strategic interest, including nanoscale material design for electronic devices and plasma control for nuclear fusion energy production. Beyond the immediate scientific advancements, the project provides broader societal benefits by supporting the training and education of three to four doctoral students at the University of Utah and the University of Wisconsin-Madison. It also funds curriculum development in data science and stochastic computation, and disseminates open-source software and research findings through international workshops to foster a highly skilled STEM workforce. This project provides a technical framework to advance the algorithmic and theoretical foundations of flow-based generative models, with a specific focus on mean flow models for scientific machine learning. While flow-based generative models learn neural network-approximated paths between prior and target distributions, they face three significant challenges: a fidelity gap compared to diffusion models, high computational complexities during sampling, and a lack of rigorous application-driven validation loops. The research goals are organized into three integrative thrusts to overcome these barriers. First, the investigators will develop partial differential equation and variational methods to analyze and minimize sampling errors, explicitly controlling Kullback-Leibler divergence and total variation distance to close the fidelity gap. Second, the project will enhance computational efficiency by extending Feynman-Kac steering guidance to mean flow models and optimizing learnable prior-target coupling schedules to construct straight flow paths that reduce function evaluations. Third, the developed algorithms will be validated using high-dimensional kinetic equations from physics and engineering. Specifically, the team will implement an offline-online sampling framework to solve Bayesian inverse problems for nanoscale heat conduction in the phonon transport equation, alongside exploring rugged control landscapes to stabilize plasma dynamics governed by the Vlasov-Poisson system. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.