– Physics-Informed, Structure-Preser ved Learning for Problems with Complex Geometries

Jian-xun Wang, University of Notre Dame< /p>

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Michael Brenner, Harvard Univer
sity

First-principle modelin g and simulation of complex systems based on partial differential equations (PDEs) and numerical discretization have been developed for decades and ac hieved great success. Nonetheless, traditional numerical solvers face signi ficant challenges in many practical scenarios, e.g., inverse problems, unce rtainty quantification, design, and optimizations. Moreover, for complex sy stems, the governing equations might not be fully known due to a lack of co mplete understanding of the underlying physics, for which a first-principle d numerical solver cannot be built. Recent advances in data science. and ma chine learning, combined with the ever-increasing availability of high-fide lity simulation and measurement data, open up new opportunities for develop ing data-enabled computational mechanics models. Although the state-of-the- art machine/deep learning techniques hold great promise, there are still ma ny challenges - e.g., requirement of “big data”, the challenge in generaliz ability/extrapolibity, lack of interpretability/explainability, etc. On the other hand, there is often a richness of prior knowledge of the systems, i ncluding physical laws and phenomenological principles, which can be levera ged in this regard. Thus, there is an urgent need for fundamentally new and transformative machine learning techniques, closely grounded in physics, t o address the aforementioned challenges in computational mechanics problems .

This talk will briefly discuss our recent developments of scientifi c machine learning for computational mechanics, focusing on several differe nt aspects of how to bake physics-induced bias into machine/deep learning m odels for data-enabled predictive modeling. Specifically, the following top ics will be covered: (1) PDE-structure preserved deep learning, where the n eural network architectures are built by preserving mathematical structures of the (partially) known governing physics for predicting spatiotemporal d ynamics, (2) physics-informed geometric deep learning for predictive modeli ng involving complex geometries and irregular domains.

Dr . Jian-xun Wang is an assistant professor of Aerospace and Mechanical Engin eering at the University of Notre Dame. He received a Ph.D. in Aerospace En gineering from Virginia Tech in 2017 and was a postdoctoral scholar at UC B erkeley before joining Notre Dame in 2018. He is a recipient of the 2021 NS F CAREER Award. His research focuses on scientific machine learning, data-e nabled computational modeling, Bayesian data assimilation, and uncertainty quantification.