WANCO: Weak Adversarial Networks for Constrained Optimization problems

TL;DR

WANCO introduces a weak adversarial network framework that converts constrained optimization problems into a minimax form via an augmented Lagrangian, representing primal variables and Lagrange multipliers with separate neural networks and training them adversarially. It demonstrates robustness to hyperparameters and effective constraint enforcement across PDE and non-PDE settings, including Ginzburg–Landau energy with mass conservation, Dirichlet/periodic partition problems, fluid–solid topology optimization, and obstacle problems, often outperforming penalty-based or plain Lagrangian approaches. The method relies on ResNet-based representations for the primal variable, tanh^3 activations, and MC-based integration, with boundary and inequality constraints incorporated through network design or auxiliary adversaries. Overall, WANCO offers a mesh-free, scalable approach for high-dimensional constrained optimization in scientific computing with broad practical impact.

Abstract

This paper focuses on integrating the networks and adversarial training into constrained optimization problems to develop a framework algorithm for constrained optimization problems. For such problems, we first transform them into minimax problems using the augmented Lagrangian method and then use two (or several) deep neural networks(DNNs) to represent the primal and dual variables respectively. The parameters in the neural networks are then trained by an adversarial process. The proposed architecture is relatively insensitive to the scale of values of different constraints when compared to penalty based deep learning methods. Through this type of training, the constraints are imposed better based on the augmented Lagrangian multipliers. Extensive examples for optimization problems with scalar constraints, nonlinear constraints, partial differential equation constraints, and inequality constraints are considered to show the capability and robustness of the proposed method, with applications ranging from Ginzburg--Landau energy minimization problems, partition problems, fluid-solid topology optimization, to obstacle problems.

Figures (18)

• Figure 1: Network structure of WANCO, see Section \ref{['Sec: Framework of WANCO']}.
• Figure 1: Comparison of the training results among WANCO, DRM-AP, and DRM-P under different $\beta$. See Section \ref{['subsection:GLcomparison']}.
• Figure 1: Results for the Dirichlet partition problem with Dirichlet boundary conditions in the unit square $[0,1]\times[0.1]$. First and third rows: the training solution $\sum_{i=1}^n(u_i(\mathbf{x};\theta))$ for $n=2$-$9$. Second and fourth rows: the corresponding partitions after the projection \ref{['formula: projection 2d n phase']} for $2$-$9$ partitions. These results are based on the evaluation of the trained neural network on $1000\times 1000$ grid points. See Section \ref{['subsec:Dirichletboundary']}.
• Figure 1: The phase variable (left), projection fluid-solid region (middle), and the velocity field (right) for Example $1$. See Section \ref{['Subsec: Fluid-solid optimization']}
• Figure 1: Training results for obstacle problems under three different obstacles and boundary conditions. The first row shows the obstacles, true solutions for the corresponding obstacles, and the training results for each case. The second row represents the point-wise error between the true solution and the trained solution. The left, middle, and right columns correspond to obstacles $\psi_1$, $\psi_2$, and $\psi_3$, respectively. See Section \ref{['Subsec: Obstacle problem']}.