PDE-GANet: Partial differential equation discovery powered by adversarial learning.

Abstract

Partial differential equations (PDEs) are typically derived through theoretical deduction or empirical observation of the real world phenomena, such as the Navier-Stokes equation and the Boltzmann equation. However, for complex systems in domains like physics, economics and meteorology, even experts encounter challenges in formulating governing PDEs. In recent years, deep neural networks have enabled automatic discovery of governing PDEs from data, positioning data-driven PDE discovery as a focal point in both academic and industrial research. Focusing on the representation and learning strategy of PDEs, we propose a bidirectional network, PDE-GANet, based on a generative adversarial network. PDE-GANet utilizes symbolic networks as the generator to represent the governing PDE expression and estimate its numerical solutions; meanwhile, it adopts a recurrent neural network as the discriminator to assess whether the estimated numerical solutions satisfy the inferred PDE from the perspective of temporal sequence. Experimental results demonstrate that PDE-GANet discovers PDEs from data with higher expression accuracy and estimates more precise numerical solutions than the state-of-the-art PDE solving methods. We believe that this work will advance the potential applications of PDEs across diverse disciplines.