A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations
This addresses the curse of dimensionality in numerical algorithms for random PDEs, which is a problem for computational scientists and engineers, though it appears incremental as it builds on existing deep learning approaches.
The authors tackled the challenge of solving high-dimensional random partial differential equations (PDEs) by developing a deep neural network surrogate framework, achieving satisfactory accuracy compared to Monte Carlo-based finite element methods on diffusion and heat conduction problems.
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.