A deep learning solution approach for high-dimensional random 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.

Weak adversarial networks for high-dimensional partial differential equations

“…Another appealing approach that exploits the variational form of PDEs is considered in [9,10,11]. In [9], a committer function is parameterized by a neural network whose weights are obtained by optimizing the variational formulation of the corresponding PDE.…”

“…In [9], a committer function is parameterized by a neural network whose weights are obtained by optimizing the variational formulation of the corresponding PDE. In [10], deep learning technique is employed to solve low-dimensional random PDEs based on both strong form and variational form. More recently, an adaptive collocation strategy is presented for a method in [27].…”

“…in Ω (20) where Ω = (−1, 1) d ⊂ R d . We first give an example of solving the IBVP (20) in dimension d = 5 using Algorithm 2 which discretizes time and uses the Crank-Nicolson scheme (10). In this test, we set f (x, t) = (π 2 − 2) sin ( π 2 x 1 ) cos( π 2 x 2 )e −t − 4 sin 2 ( π 2 x 1 ) cos( π 2 x 2 )e −2t in Ω × [0, T ], g(x, t) = 2 sin( π 2 x 1 ) cos( π 2 x 2 )e −t on ∂Ω × [0, T ] and h(x) = 2 sin( π 2 x 1 ) cos( π 2 x 2 ) in Ω.…”

“…Solving general high-dimensional partial differential equations (PDEs) has been a long-standing challenge in numerical analysis and computation [1,2,3,4,5,6,7,8,9,10,11,12]. In this paper, we present a novel method that leverages the form of weak solutions and adversarial networks to compute solutions of PDEs, especially to tackle problems posed in high dimensions.…”

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

“…Additionally, Largaris et al [14] and more recently Berg and Nyström [16] showed fully connected networks can be used to learn PDE solutions on even complex domains. Recently, several investigators have examined the use of variational formulations of the governing equations as loss functions to solve various PDEs [3,17,18,19] which has been proven to be effective. Sirignano et al [20] show that the use of a fully connected network can be used for efficiently solving PDEs of high dimensionality where traditional discretization techniques become unfeasible.…”