Deep convolutional Ritz method: parametric PDE surrogates without labeled data

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