CAREER: Theoretically Guaranteed Structure-Informed Machine Learning and Data-Driven Structural Inference

Machine learning is rapidly changing science, engineering, and technology, but many of its most successful methods require enormous amounts of high-quality data. This limits their usefulness in scientific settings where data may be expensive, noisy, incomplete, or difficult to obtain, such as turbulence modeling, molecular design, medical imaging, and the study of complex physical systems. This project will develop new mathematical foundations and computational tools that allow machine learning systems to make better use of limited data by incorporating structure, such as symmetry, low-dimensional patterns, and physical constraints. It will also develop methods for discovering hidden structure directly from data, including conservation laws and governing principles of dynamical systems. These advances will support more reliable, interpretable, and efficient artificial intelligence for scientific discovery, aligning with national priorities in artificial intelligence and strengthening the mathematical foundations needed for future scientific and engineering applications. The project will also contribute to education and workforce development through new courses, undergraduate research opportunities, data science bootcamps, open-source software, and outreach activities for students in grades 8 through 12, thereby broadening participation and preparing students for careers at the interface of mathematics, computing, and science. The investigator will study the two-way relationship between data and structure in machine learning. One direction will develop structure-informed learning models that incorporate known structural information, with emphasis on generative models and diffusion models that preserve symmetries, multimodality, low-dimensional structure, and other physically meaningful priors. The project will establish rigorous sample-complexity and computational-complexity theory to quantify when such structure improves learning, when the benefit saturates, and how structural priors interact with optimization and regularization. A second direction will develop data-driven methods for discovering unknown structure in dynamical systems, including conservation laws, integrability, and Lax pairs. These methods will combine neural-network-based learning, deflation strategies, symbolic regression, and theoretical analysis to produce interpretable models and principled guarantees. The expected outcomes include new mathematical theory, scalable algorithms, benchmark problems, open-source software, and practical guidance for designing reliable structure-informed artificial intelligence systems for scientific computing and 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. NSF Award ID: 2540370 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT,01003031DB NSF RESEARCH & RELATED ACTIVIT,01002930DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Wei Zhu | Institution: Georgia Tech Research Corporation, ATLANTA, GA | Award Amount: $246,901 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2540370 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2540370.html