  Parametric surrogate models for partial differential equations (PDEs) are a
necessary component for many applications in the computational sciences, and
convolutional neural networks (CNNs) have proved as 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 convolutional physics-informed
neural network (CPINN) 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 found that CNNs
trained on labeled data outperform CPINNs in convergence speed and
generalization ability. Surrogates generated from DCRM, however, converge
significantly faster than their CPINN counterparts and prove to generalize
faster and better than surrogates obtained from both CNNs trained on labeled
data and CPINNs. This hints that DCRM could make PDE solution surrogates
trained without labeled data possible.
