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.

利用深度学习方法解决高维随机偏微分方程的问题。通过使用全连接深度残差网络来逼近随机偏微分方程，在确定逼近深度神经网络的参数时，采用了 SGD 的变种，并在扩散和热传导问题上得到了验证。