Mathematical Foundations and Scalable Algorithms for High-Dimensional Sampling and Generative Modeling

Artificial intelligence and scientific computing increasingly rely on mathematical models of uncertainty. A central task is to generate representative possible outcomes from these models, a process known as sampling. In scientific inference, the model combines scientific knowledge with observations, while in generative artificial intelligence, it is learned from data. Modern problems may involve enormous numbers of variables or highly detailed data. As their size or resolution grows, current methods can become too slow, and generated data can lose small-scale features that matter scientifically. This project develops faster and more reliable methods for these high-dimensional problems, with applications such as interpreting astronomical observations, measuring uncertainty in scientific models, and generating turbulent-flow data. By strengthening probabilistic computation, the project promotes the progress of science, supports United States leadership in science and technology, and contributes to national prosperity. The work also supports Presidential priorities in artificial intelligence by strengthening the mathematical foundations of generative artificial intelligence for science. The project's broader impacts include the release of open source software, integration of project ideas into interdisciplinary courses and freely available educational materials, and research opportunities for students from precollege through graduate study. These activities will broaden participation in mathematical and computational science and strengthen the national scientific workforce. The project develops a common mathematical and computational framework for sampling and generative modeling in high dimensions. The investigator studies three connected research directions. The first determines when numerical errors in low-dimensional marginals remain small even as error in the full distribution grows with dimension. The second addresses ill-conditioning that can worsen with dimension by developing affine-invariant Hamiltonian samplers and derivative-free ensemble Kalman methods. The third treats generative modeling of multiscale scientific data as a numerical analysis problem and uses Lipschitz regularity to improve conditioning and fine-scale accuracy as resolution grows. Expected contributions include new theory for high-dimensional sampling, open source sampling algorithms for Bayesian inference, and better-conditioned generative methods for scientific data. 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.