Computational Mathematics and Scientific Computing Seminar

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As scientific machine learning and reduced order models begin to offer forward simulation tools fast enough for near-real time prediction, many have started developing "digital twins" - frameworks which perform real time prediction and data-assimilation in a closed loop to continually update a model of a system. The application of these models to high consequence science and engineering settings however requires notions of numerical robustness and trust that current ML approaches generally lack: numerical stability, physical realizability, explainability, causal mechanistic relationships, and uncertainty quantification are all needed. In this talk, we present our work from the last few years developing a finite element exterior calculus which allows one to learn physics models from data while maintaining the theoretical guarantees of mixed finite element methods. Notably, we will show how graph attention networks admit interpretation in the finite element exterior calculus. Machine learnable Whitney forms provide a fully differentiable alternative to a traditional computational mesh, allowing the learning of physics as unknown nonlinear fluxes associated with a hypothesized conservation law, guaranteeing preservation of conservation structure as well as dissipative bracket structures. We show our recent work developing conditional extensions of the architecture. Much like how Dall-E allows sampling from the space of images with probability distribution conditioned on a text prompt, the architecture allows one to sample from the space of finite element models conditioned on experimental measurements.