Collaborative Research: MFAI: Mathematical Foundations of Score-based Generative Models

This project advances the mathematical foundations of modern generative artificial intelligence, with the goal of making these technologies more reliable, trustworthy, and effective for scientific discovery and engineering innovation. Modern generative artificial intelligence (AI) systems can create images, text, molecular structures, and scientific data with striking realism, but their success raises fundamental questions: When do these models truly generate new information rather than memorize their training data? Why can they work in very high-dimensional settings where classical methods often fail? How can they be used safely and efficiently for scientific discovery and decision-making? This project addresses these questions for score-based generative models, a leading class of diffusion-based AI methods. The results will help build a deeper understanding of when generative models generalize beyond their training data and how they can be used for conditional generation and sampling in complex scientific problems. The mathematical tools developed in this project have potential relevance to biotechnology, materials discovery, quantum science, and other areas that depend on high-dimensional modeling, uncertainty quantification, and efficient numerical simulation. The project will support interdisciplinary training for students and postdoctoral researchers, new course materials on the mathematical foundations of generative AI, research mentoring, workshops, and outreach activities designed to engage trainees at the intersection of mathematics, data science, and AI. The technical goal of the project is to establish a rigorous mathematical theory for score-based generative models, with emphasis on generalization, intrinsic structure, and accelerated sampling. The research is organized around three integrated thrusts. First, the project develops a novelty-informed theory of generalization for score-based generative models by quantifying not only the accuracy of the output distributions but also the dependence of generated samples on the training data. The project will use information-theoretic tools such as mutual information to characterize memorization, novelty, and the trade-off between them, and will analyze these questions for a suite of score estimators including those arising from kernel methods, denoising score matching, and neural-network training. Second, the project will study score-based generative models when data lie on or near low-dimensional manifolds embedded in high-dimensional spaces. This part will develop end-to-end error bounds that combine score-estimation error and reverse-time sampling error, with rates depending on intrinsic rather than ambient dimension, and will investigate how manifold-learning tools such as diffusion maps can improve score estimation and sampling. Finally, the project will develop score-based methods for accelerated conditional and posterior sampling. It will analyze guidance models, particle and birth-death sampling schemes, and deterministic flow-map approaches for using learned score-based priors in Bayesian inference and other conditional sampling problems. These integrated efforts will produce new theory for understanding and quantifying memorization in score-based generative models, new provable algorithms for structured high-dimensional data, and principled sampling methods that accelerate scientific discovery. 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.