Speaker: Nat Trask, Univ of Penn
Title: Structure-preserving generative scientific machine learning with the finite element exterior calculus
Abstract: Motivated by the ever-increasing success of machine learning in language and vision models, many aim to build AI-driven tools for scientific simulation and discovery. Contemporary techniques drastically lag behind their comparatively mature counterparts in modeling and simulation however, lacking rigorous notions of convergence, physical realizability, uncertainty quantification, and verification+validation that underpin prediction in high-consequence engineering settings. One reason for this is the use of "off-the-shelf" ML architectures designed for language/vision without specialization to scientific computing tasks. In this work, we establish connections between graph neural networks and the finite element exterior calculus (FEEC). FEEC forms the backbone of modern mixed finite element methods, tying the discrete topology of geometric descriptions of space (cells, faces, edges, nodes and their connectivity) to the algebraic structure of conservations laws (the div/grad/curl theorems of vector calculus). By building a differentiable learning architecture mirroring the construction of Whitney forms, we obtain a de Rham complex supporting FEEC, allowing us to learn models combining the robustness of traditional FEM with the drastic speedups and data assimilation capabilities of ML. We then introduce a novel UQ framework based on optimal recovery in reproducing Hilbert spaces, allowing the model to quantify epistemic uncertainty, providing practical notions of trust where the model may be reliably employed. Finally, we present an architecture we have recently developed which admits conditional generative modeling, allowing one to sample from the space of finite element models consistent with given observational data in near real time.
Bio: Dr. Trask is an associate professor in Mechanical Engineering and Applied Mechanics at the University of Pennsylvania, holds joint listings in Material Science and Applied Mathematics and Computational Science, and maintains a joint faculty appointment with Sandia National Laboratories. Prior to coming to UPenn in 2023, he was senior staff at Sandia for 8 years. He obtained his PhD from the Division of Applied Mathematics in 2016 working with Dr. Martin Maxey. Dr. Trask is the recipient of the Department of Energy Early Career Award and the NSF MSPRF award, and is deputy director of the multi-institutional DOE MMICCs center SEA-CROGS, developing ML-enabled digital twins for earth and embedded systems. His research spans a broad range of multiphysics and multiscale problems, including: fusion power, shock physics, climate, semiconductor physics, and energy storage.
In person attendance at this seminar is only open to Columbia Univesity affiliates. External guests are welcome to attend remotely. Please contact [email protected] if you need the Zoom link for this seminar.