Inverse Estimation of Elastic Modulus Using Physics-Informed Generative Adversarial Networks

TL;DR

This paper introduces physics-informed GANs (PI-GANs) that incorporate physical laws via PDEs to perform inverse estimation of spatially-varying elastic modulus from deformation data, enabling realistic probabilistic modeling with limited data.

Contribution

The work demonstrates the application of PI-GANs for elastic modulus estimation, integrating PDE constraints into GANs to infer material properties from limited deformation measurements.

Findings

PI-GANs accurately estimate the distribution of elastic modulus.

The method generates realistic, physically consistent material stiffness realizations.

Good agreement with true distribution statistics is achieved.

Abstract

While standard generative adversarial networks (GANs) rely solely on training data to learn unknown probability distributions, physics-informed GANs (PI-GANs) encode physical laws in the form of stochastic partial differential equations (PDEs) using auto differentiation. By relating observed data to unobserved quantities of interest through PDEs, PI-GANs allow for the estimation of underlying probability distributions without their direct measurement (i.e. inverse problems). The scalable nature of GANs allows high-dimensional, spatially-dependent probability distributions (i.e., random fields) to be inferred, while incorporating prior information through PDEs allows the training datasets to be relatively small. In this work, PI-GANs are demonstrated for the application of elastic modulus estimation in mechanical testing. Given measured deformation data, the underlying probability distribution of spatially-varying elastic modulus (stiffness) is learned. Two feed-forward deep neural network generators are used to model the deformation and material stiffness across a two dimensional domain. Wasserstein GANs with gradient penalty are employed for enhanced stability. In the absence of explicit training data, it is demonstrated that the PI-GAN learns to generate realistic, physically-admissible realizations of material stiffness by incorporating the PDE that relates it to the measured deformation. It is shown that the statistics (mean, standard deviation, point-wise distributions, correlation length) of these generated stiffness samples have good agreement with the true distribution.