Title: A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations,
Abstract: Developing efficient numerical algorithms for the solution of high
dimensional random Partial Differential Equations (PDEs) has been a challenging
task due to the well-known curse of dimensionality. We present a new solution
framework for these problems based on a deep learning approach. Specifically,
the random PDE is approximated by a feed-forward fully-connected deep residual
network, with either strong or weak enforcement of initial and boundary
constraints. The framework is mesh-free, and can handle irregular computational
domains. Parameters of the approximating deep neural network are determined
iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm.
The satisfactory accuracy of the proposed frameworks is numerically
demonstrated on diffusion and heat conduction problems, in comparison with the
converged Monte Carlo-based finite element results.