Stochastics for generative modeling and applications to inverse problems

Modern society increasingly relies on technologies that use waves to see inside complex environments that cannot be observed directly. Medical imaging, seismic exploration, remote sensing, and astronomical imaging all depend on interpreting how waves propagate through complicated materials in order to detect hidden structures or abnormalities. This project will develop new mathematical and artificial intelligence (AI) methodologies that will make such imaging technologies more accurate, reliable, and computationally efficient. A major focus will be on biomedical imaging, where the methods will help identify malignant tissue and improve early diagnosis of disease. The project will establish a rigorous mathematical foundation for emerging generative AI techniques used in imaging and uncertainty quantification, thereby improving confidence in AI-assisted scientific and medical decision making. The research will also strengthen connections between mathematics, physics, engineering, and data science, while training graduate and undergraduate students in interdisciplinary research areas of growing societal importance. The project will develop new mathematical theory for Bayesian inverse problems and generative modeling in infinite-dimensional settings arising from wave propagation in complex media. The research will analyze stochastic differential equations, their time reversal, and associated sampling dynamics in order to construct efficient algorithms that rapidly approach conditional probability distributions informed by observational data and prior information. The work will incorporate optimal annealing strategies, preconditioning methods, and generalized Lévy driving processes to improve sampling efficiency and robustness. A particular emphasis will be placed on Magnetic Resonance Elastography and related tissue imaging techniques, where dispersive elastic wave effects will be exploited to infer tissue morphology and detect pathological structures. The project will further mathematical theory for waves in random media and develop computational tools for rapid conditional sampling and generative AI more broadly. The resulting techniques will have applications extending beyond medical imaging to remote sensing, reflection seismology, and other imaging sciences. The project will also support a seminar series and a Southern California inverse problems workshop, foster collaborations across applied mathematics, biomedical engineering, and physics, and provide research opportunities for graduate and undergraduate students. The biomedical imaging advances developed in the project will also create strong opportunities for future technological translation and commercialization. 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. NSF Award ID: 2606808 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Knut Solna | Institution: University of California-Irvine, IRVINE, CA | Award Amount: $209,660 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2606808 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2606808.html