Weak adversarial networks for high-dimensional partial differential equations

Abstract

Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.

A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions

“…In the past decade, deep learning has achieved great success in many subjects, like computer vision, speech recognition, and natural language processing [7,11,18] due to the strong representability of deep neural networks (DNNs). Meanwhile, DNNs have also been used to solve partial differential equations (PDEs); see for example [1,4,5,8,19,22,24,26,28]. In classical numerical methods such as finite difference method [20] and finite element method [2], the number of degrees of freedoms (dofs) grows exponentially fast as the dimension of PDE increases.…”

“…Afterwards, Monte-Carlo method is used to approximate the loss (objective) function which is defined over a high-dimensional space. Some methods are based on the PDE itself [24,26] and some other methods are based on the variational or the weak formulation [5,16,21,28]. Another successful example is the multilevel Picard approximation which is provable to overcome the curse of dimensionality for a class of semilinear parabolic equations [13].…”

A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems

“…In recent years, neural networks have shown great success in representing highdimensional classifiers or probability distributions in a variety of machine learning tasks and have led to the tremendous success and popularity of deep learning [32,38]. Motivated by those recent success, researchers have been actively exploring using deep learning techniques to solve high dimensional PDEs [10,18,22,23,29,37,45,48] by using neural networks to parameterize the unknown solution of high dimensional PDEs. Thanks to the flexibility of the neural network approximations, such methods have achieved remarkable results for various kind of PDE problems, including eigenvalue problems for many-body quantum systems (see e.g., [7,9,12,19,[24][25][26]36]), where the high dimensional wave functions are parameterized by neural networks with specific architecture design to address the symmetry properties of many-body quantum systems.…”

A Local Deep Learning Method for Solving High Order Partial Differential Equations

“…[21,30] provides multi-scale deep neural network methods which separate different frequencies of the loss function and approximate them by the corresponding neural networks. [37] considers variational problems, and the loss function is defined as a weak formulation. To deal with the essential boundary conditions, [20] resorts to Nitsche's variational formulation.…”

“…Addressing the issue of deep learning methods, [29] proposes a Monte Carlo method to approximate second order derivatives. In [32,37], the variational form reduces the order of derivatives through the integration by parts. [11] proposes a derivative free method for parabolic PDEs by solving the equivalent BSDE problem.…”