Weak Adversarial Networks for High-dimensional Partial Differential Equations

By Yaohua Zang, Gang Bao, Xiaojing Ye, Haomin Zhou. Journal of Computational Physics, 2019

537 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/80458b2466d4a665fd22cb9c795d2a4350e0efba

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. G. Em Karniadakis
2. E. Kharazmi
3. Zhongqiang Zhang
4. Julius Martensen
5. K. Zubov
6. Collin Warner
7. Dominic J. Skinner
8. Yingbo Ma

In Statistical and Nonlinear Physics

1. Zhongqiang Zhang
2. G. Em Karniadakis
3. E. Kharazmi
4. Julius Martensen
5. K. Zubov
6. Collin Warner
7. Dominic J. Skinner
8. Yingbo Ma

In Statistics and Probability

1. Zhongqiang Zhang
2. E. Kharazmi
3. G. Em Karniadakis
4. Dongbin Xiu
5. Anthony Nouy
6. Nicholas Zabaras
7. Xiaoliang Wan
8. Youssef M. Marzouk

Citations in the corpus

• PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, unknown): “, strong/variational formulations of PDE residuals [45–48], distributed learning using domain decomposition [49–51], and convergence analysis in network training/optimizations [41, 52, 53].”
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition (2020, background): “2) into a system of first-order equations, and WAN [34, 35] develops an adversarial framework by taking the test function to be a separate network.”
• Universal Differential Equations for Scientific Machine Learning (2020, background): “…equations into alternative forms, such as backwards stochastic differ-ential equations (BSDEs), which are then approximated by neural networks have been shown to be highly efficient on important equations such as the nonlinear Black-Scholes and Hamilton-Jacobi-Bellman (HJB) equations [91, 92, 93, 94].”