Weak Minimization-Sampling Network for Solving PDEs

Abstract

In many engineering applications, people concentrate on deriving the weak solution to a given partial differential equation (PDE) model. Traditionally, both the trial space and the test space are constructed as the span of a set of basis functions with compact supports and the approximate solution is achieved by solving a (linear) system of equations. A recent advancement in this field is the Weak Adversarial Network (WAN), a machine learning-based method that employs deep neural networks (DNNs) to approximate trial and test spaces. WAN addresses PDEs through a min-max optimization framework, showcasing its effectiveness in solving high dimensional PDEs and surpassing the limitations of traditional numerical methods. In this paper, motivated by WAN, we propose a Weak Minimization-Sampling Network (WMSN) for solving PDEs which also leverages DNNs to approximate the trial and test spaces. Instead of the min-max loss function, our key idea is to reformulate the weak formulation into a loss function defined as an expectation over all possible DNNs in the test space with only the network parameters in the trial space to be trained. This leads to a minimization problem in the trial space and a sampling problem over the test space, which can be solved by stochastic gradient descent type methods with neat implementation. Through a series of numerical tests, we find that WMSN exhibits comparable or slightly better results with smaller computational cost when compared to WAN.