Collaborative Research: Geometric Scientific Machine Learning for PDEs with Tensorial Constraints

As artificial intelligence (AI) increasingly accelerates scientific discovery and engineering design, there is a growing need for models that are not only computationally fast but physically reliable. Many current AI approaches rely purely on massive datasets, predicting physical phenomena without incorporating the underlying laws of nature. This purely data-driven approach can lead to predictions that are unstable or physically impossible. This project develops physics-preserving machine learning models that embed geometric and physical constraints directly into the AI's architecture. By ensuring these models obey fundamental physical laws by design, the research yields simulators that operate thousands of times faster than traditional computational methods without sacrificing accuracy. These advancements directly support Federal strategic interests in artificial intelligence and advanced manufacturing by enabling the creation of real-time, highly accurate digital twins for complex systems in aerospace, materials science, and energy. Additionally, the project supports workforce development by training a new generation of scientists, spanning high school, undergraduate, and graduate levels at the critical intersection of computational mathematics and machine learning. This project will create structure-preserving scientific machine learning (SciML) architectures to learn reduced partial differential equation (PDE) models incorporating constrained tensors. Examples include the stress and strain tensors in linear elasticity (symmetric), deviatoric stress (trace-free), and internal stress or linearized Riemann curvature (antisymmetric in pairs, pair exchange symmetry, algebraic Bianchi identities). The PDEs and tensor constraints will be formulated using higher-order differential complexes created by combining de Rham complexes via the Bernstein-Gelfand-Gelfand (BGG) technique. This ensures data-driven surrogates inherit the geometric structure inherent in the tensor objects. Rather than treating physics as data-driven regression, the approach performs learning at the level of the physics to build transformer models thousands of times faster than forward simulators, with geometric structure inherited from the BGG complexes providing guarantees of trustworthy performance. This work will lay the foundation for a unified theory of data-driven physics matching the rigor of finite element methods while preserving the approximation power of modern transformer methods. 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: 2608775 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Anil Hirani | Institution: University of Illinois at Urbana-Champaign, URBANA, IL | Award Amount: $349,985 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2608775 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2608775.html