ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains

Highlights: • Solutions to PDEs on general domains are accurately approximated by a single forward-pass of a convolutional neural network. • A theoretical justification for the neural network approximation is established based on the theory of Green's functions. • Neural networks are trained to produce accurate uncertainty estimates using a scalable framework suitable for large datasets. • The empirical statistics of the uncertainty estimates are observed to align precisely with the designed Gaussian error model. -- Abstract: In this work, we introduce the ConvPDE-UQ framework for constructing light-weight numerical solvers for partial differential equations (PDEs) using convolutional neural networks. A theoretical justification for the neural network approximation to partial differential equation solvers on varied domains is established based on the existence and properties of Green's functions. These solvers are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a convolutional network. The network architecture is also designed to predict pointwise Gaussian posterior distributions, with weights trained to minimize the associated negative log-likelihood of the observed solutions. This setup facilitates simultaneous training and uncertainty quantification for the network's solutions, allowing the solver to provide pointwise uncertainties for its predictions. The associated training procedure avoids the computationally expensive Bayesian inference steps used by other state-of-the-art uncertainty models and allows training to be scaled to the large data sets required for learning on varied problem domains. The performance of the framework is demonstrated on three distinct classes of PDEs consisting of two linear elliptic problem setups and a nonlinear Poisson problem. After a single offline training procedure for each class, the proposed networks are capable of accurately predicting the solutions to linear and nonlinear elliptic problems with heterogeneous source terms defined on any specified two-dimensional domain using just a single forward-pass of a convolutional neural network. Additionally, an analysis of the predicted pointwise uncertainties is presented with experimental evidence establishing the validity of the network's uncertainty quantification schema.