MGDGAN: Multiple Generator and Discriminator Generative Adversarial Networks for Solving Stochastic Partial Differential Equations

Abstract

We propose novel structures of generator and discriminator in physics-informed generative adversarial networks called multiple-generator-and-discriminator generative adversarial networks (MGDGANs), that are designed to solve stochastic partial differential equations (SPDEs). MGDGANs for SPDEs consist of three steps: a generator that samples a solution to the SPDEs, a physics-informed operator that enforces the governing equation, and a discriminator that distinguishes between samples from the generator and training samples. Inspired by the polynomial chaos, we represent the solution by the inner product of functions in spatial and random variables, and model each function by a separate generator. We show that the proposed multiple generator structure offers huge computational savings in training and prediction. If multiple stochastic processes exist in the system, then a distinct discriminator is used for each of them. We show that the loss function obtained by these distinct discriminators provides an equivalent metric to the Wasserstein distance loss by a single discriminator, and provide numerical examples to demonstrate that these multiple discriminators enhance the training accuracy. Numerical examples are demonstrated to verify that the proposed model is efficient in computation and memory; the model reduces computing time by more than a factor of 10 and relative l2 error by about one-third in the SPDE example.

Keywords

• Computational modeling
• deep learning
• generative adversarial networks
• physics-informed deep generative models
• uncertainty quantification