Applied & Computational Mathematics seminar: Structure preserving digital twins via conditional neural Whitney forms

Speaker: Nat Trask, University of Pennsylvania

Title:  Structure preserving digital twins via conditional neural Whitney forms

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 are able to learn models combining the robustness and UQ of traditional FEM with the drastic speedups and data assimilation capabilities of ML. 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.

Time:  Friday, February 28 – 10:00am – 11:00am

Place:  Exploratory Hall, room 4106 and Zoom