PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations

TL;DR

PI-VEGAN is a novel physics-informed neural network framework that combines variational embedding and adversarial training to solve stochastic differential equations efficiently, accurately, and with limited data.

Contribution

The paper introduces PI-VEGAN, integrating variational encoding with physics-informed GANs to handle forward, inverse, and mixed stochastic differential equation problems.

Findings

Achieves high stability and accuracy in solving stochastic PDEs.

Effectively incorporates physical laws via automatic differentiation.

Outperforms previous physics-informed GANs in numerical experiments.

Abstract

We present a new category of physics-informed neural networks called physics informed variational embedding generative adversarial network (PI-VEGAN), that effectively tackles the forward, inverse, and mixed problems of stochastic differential equations. In these scenarios, the governing equations are known, but only a limited number of sensor measurements of the system parameters are available. We integrate the governing physical laws into PI-VEGAN with automatic differentiation, while introducing a variational encoder for approximating the latent variables of the actual distribution of the measurements. These latent variables are integrated into the generator to facilitate accurate learning of the characteristics of the stochastic partial equations. Our model consists of three components, namely the encoder, generator, and discriminator, each of which is updated alternatively employing the stochastic gradient descent algorithm. We evaluate the effectiveness of PI-VEGAN in addressing forward, inverse, and mixed problems that require the concurrent calculation of system parameters and solutions. Numerical results demonstrate that the proposed method achieves satisfactory stability and accuracy in comparison with the previous physics-informed generative adversarial network (PI-WGAN).