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

arXiv: https://arxiv.org/abs/2006.05791 · HTML: https://mlsanity.com/paper/2006.05791/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2020.

## Built on by (most cited first, 4 of 4)

- [Physics-Informed Deep Learning: A Promising Technique for System Reliability Assessment](https://mlsanity.com/paper/2108.10828/) (2021)
- [Latent Generative Modeling of Random Fields from Limited Training Data](https://mlsanity.com/paper/2505.13007/) (2025)
- [GenVoid: Uncertainty-Aware Learning of Subsurface Material Defects with an Experimentally Validated Physics-Informed Generative Model](https://mlsanity.com/paper/2609.23761/) (2026)
- [Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems](https://mlsanity.com/paper/2603.29515/) (2026)

## Abstract (arXiv)

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.