Session: 13-19-02: Scientific Machine Learning (SciML) for Characterization, Modeling, and Design of Structures and Materials II

Paper Number: 173923

Diffusion Models for Effective Modeling of High-Frequency Waves and Microstructure in Mechanics 

Modeling high frequency waves in elastodynamics and non-convex variational problems are ubiquitous family of problems in mechanics that both share the emergence of fine microstructure in solutions. The former arises in elastodynamics and modeling of high frequency dilatational waves or in modeling of shear waves, while the latter arises in modeling microstructure of materials. Both also emerge in homogenization problems.  Traditional methods like Finite Element Method (FEM) face canonical issues like mesh-dependency in solving such problems. Yet, we often are interested in some average measures of the solutions rather than the point-wise solution.

We have developed novel frameworks using scientific machine learning (SciML) to learn such average quantities of interest. For the non-convex variational problems, after discussing emergence of measures in inelastic problems, we present our framework for neural network representation of Young measures. We will learn the Young measures by incorporating the variational problem in the loss function, and employing a ResNet architecture. This is achieved by representing Young measures as a push-forward map of a Gaussian measures. We demonstrate our framework by solving several numerical problems, starting with the famous Bolza problem in one-dimensional calculus of variations. We will further discuss the applications of our framework for nonlinear mechanics and homogenization, and how it enables bypassing the tedious task of quasi-convexification and directly approximating the effective solution.

In the second half of the talk, we dive into the modeling of high frequency waves in steady state elastodynamics. Observing that prediction of waves in high frequency far from the source field is inherently a difficult task, we ask the following question: while learning the correct far field wave patterns and phases is very challenging, can one learn some quantities of interest like energy accurately? We will demonstrate that adopting a probabilistic machine learning framework enables such predictions. In particular, we demonstrate diffusion model capabilities in predicting solutions of Helmholtz equations from the knowledge of sound speed map. We demonstrate their accuracy and benchmark them with other neural operator techniques. We further delineate the versatility of diffusion models in quantifying input uncertainties.

After showcasing two classes of problems in solid mechanics that scientific machine learning truly makes a difference in juxtaposition with traditional computational capabilities such as FEM, we conclude by highlighting some nuances of resorting to SciML neural network representations for solving problems in mechanics. We highlight some restrictions of these methods, as well as discussing some future pathways.

Presenting Author: Yicheng Zou Duke University

Presenting Author Biography: Professor (Amir)Hossein Salahshoor is an Assistant Professor in the departments of Civil and Environmental Engineering, and Mechanical Engineering and Material Science at Duke. Before that, he conducted his postdoctoral studies at Caltech. Prior to that he obtained his Ph.D in Aerospace Engineering from Georgia Tech, along with an MS in Mathematics. His research interests broadly lie at the intersection of mechanics of materials and structures, computational and data science, biology, and applied mathematics.

Authors:

Yicheng Zou Duke University
Hossein Salahshoor N/A