A deep learning solution approach for high-dimensional random differential equations

• Published in: Probabilistic engineering mechanics Vol. 57; pp. 14 - 25
• Main Authors: Nabian, Mohammad Amin, Meidani, Hadi
• Format: Journal Article
• Language: English
• Published: Barking Elsevier Ltd 01.07.2019; Elsevier Science Ltd
• Subjects: Algorithms; Artificial neural networks; Conduction heating; Conductive heat transfer; Curse of dimensionality; Deep learning; Deep neural networks; Finite element analysis; Least squares; Meshless methods; Monte Carlo simulation; Neural networks; Partial differential equations; Random differential equations; Residual networks; Stochastic models; Deep learning; Residual networks; Curse of dimensionality; Deep neural networks; Least squares; Random differential equations
• ISSN: 0266-8920; 1878-4275
• DOI: 10.1016/j.probengmech.2019.05.001

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 approach for these problems based on deep learning. This approach is intrusive, entirely unsupervised, and mesh-free. 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. 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 approach is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.