Past Event: Oden Institute Seminar

Learning Nonlocal Neural Operators for Complex Material Modeling

Prof. Yue Yu, Department of Mathematics, Lehigh University

Abstract

Over the past several decades, physics-based Partial Differential Equations (PDEs) have been the cornerstone for modeling complex material responses. Traditional numerical methods have long been employed to solve these PDEs and provide predictions. However, their accuracy and computational feasibility can be compromised when dealing with unknown governing laws or high levels of material heterogeneity. In this talk, we propose to use data-driven modeling approaches which directly utilize high-fidelity simulation and/or experimental measurements to learn the hidden physics and provide further predictions. In particular, we develop nonlocal neural operator architectures, to learn the mapping between loading conditions and the corresponding material responses. By parameterizing the increment between layers as an integral operator, our neural operator can be seen as an analog of a time-dependent nonlocal equation. This framework adeptly captures long-range dependencies within the feature space, ensuring applicability across various resolutions. As such, our neural operator acts as an efficient surrogate PDE solver and provides a universal approximator to a fixed point iterative procedure. Moreover, the nonlocal architecture can be customized to uphold fundamental physical laws such as the conservation of mass and linear/angular momentum. This feature substantially enhances the model's generalizability and reliability. To illustrate the real-world applicability of our approach, we demonstrate the direct learning of material models from digital image correlation (DIC) displacement tracking measurements on porcine tricuspid valve leaflet tissues. Our results showcase the superior performance of the learned model when compared to traditional constitutive models.

Biography

Yue Yu received her B.S. from Peking University in 2008, and her Ph.D. from Brown University in 2014. She was a postdoc fellow at Harvard University after graduation, and then she joined Lehigh University as an assistant professor of applied mathematics and was promoted to full professor in 2023. Her research lies in the area of applied mathematics and computational mechanics, with recent projects focusing on nonlocal problems and scientific machine learning. She has received an NSF Early Career award and an AFOSR Young Investigator Program (YIP) award.