A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks

TL;DR

This paper introduces a novel framework using conditional GANs to learn forward and inverse PDE solutions, specifically for complex heterogeneous porous media, achieving high accuracy and efficiency.

Contribution

It adapts image-to-image translation with cGANs for PDE solution operators, addressing heterogeneity and discontinuities in complex media.

Findings

Competitive accuracy with state-of-the-art methods

Reduced computational and training time

Effective for both forward and inverse PDE problems

Abstract

This work is the first to employ and adapt the image-to-image translation concept based on conditional generative adversarial networks (cGAN) towards learning a forward and an inverse solution operator of partial differential equations (PDEs). Even though the proposed framework could be applied as a surrogate model for the solution of any PDEs, here we focus on steady-state solutions of coupled hydro-mechanical processes in heterogeneous porous media. Strongly heterogeneous material properties, which translate to the heterogeneity of coefficients of the PDEs and discontinuous features in the solutions, require specialized techniques for the forward and inverse solution of these problems. Additionally, parametrization of the spatially heterogeneous coefficients is excessively difficult by using standard reduced order modeling techniques. In this work, we overcome these challenges by employing the image-to-image translation concept to learn the forward and inverse solution operators and utilize a U-Net generator and a patch-based discriminator. Our results show that the proposed data-driven reduced order model has competitive predictive performance capabilities in accuracy and computational efficiency as well as training time requirements compared to state-of-the-art data-driven methods for both forward and inverse problems.