Deep convolutional Ritz method: parametric PDE surrogates without labeled data

Abstract

The parametric surrogate models for partial differential equations (PDEs) are a necessary component for many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool to generate these surrogates when parametric fields are present. CNNs are commonly trained on labeled data based on one-to-one sets of parameter-input and PDE-output fields. Recently, residual-based deep convolutional physics-informed neural network (DCPINN) solvers for parametric PDEs have been proposed to build surrogates without the need for labeled data. These allow for the generation of surrogates without an expensive offline-phase. In this work, we present an alternative formulation termed deep convolutional Ritz method (DCRM) as a parametric PDE solver. The approach is based on the minimization of energy functionals, which lowers the order of the differential operators compared to residual-based methods. Based on studies involving the Poisson equation with a spatially parameterized source term and boundary conditions, we find that CNNs trained on labeled data outperform DCPINNs in convergence speed and generalization abilities. The surrogates generated from the DCRM, however, converge significantly faster than their DCPINN counterparts, and prove to generalize faster and better than the surrogates obtained from both CNNs trained on labeled data and DCPINNs. This hints that the DCRM could make PDE solution surrogates trained without labeled data possibly.