# WANCO: Weak Adversarial Networks for Constrained Optimization problems

arXiv: https://arxiv.org/abs/2407.03647 · HTML: https://mlsanity.com/paper/2407.03647/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 52 references that are in the map.
- First posted on arXiv 4 Jul 2024.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [Constrained Optimization to Train Neural Networks on Critical and Under-Represented Classes](https://mlsanity.com/paper/2102.12894/) (2021)
- [An augmented Lagrangian deep learning method for variational problems with essential boundary conditions](https://mlsanity.com/paper/2106.14348/) (2021)
- [Lagrangian dual framework for conservative neural network solutions of kinetic equations](https://mlsanity.com/paper/2106.12147/) (2021)
- [A physics-informed neural network framework for modeling obstacle-related equations](https://mlsanity.com/paper/2304.03552/) (2023)
- [ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.12433/) (2023)
- [An efficient threshold dynamics method for topology optimization for fluids](https://mlsanity.com/paper/1812.09437/) (2018)
- [Numerical approximation of phase field based shape and topology optimization for fluids](https://mlsanity.com/paper/1405.3480/) (2014)
- [A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results](https://mlsanity.com/paper/1606.02513/) (2016)
- [A diffusion generated method for computing Dirichlet partitions](https://mlsanity.com/paper/1802.02682/) (2018)
- [An efficient unconditionally stable method for Dirichlet partitions in arbitrary domains](https://mlsanity.com/paper/2108.11552/) (2021)

## Built on by (most cited first, 3 of 3)

- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning](https://mlsanity.com/paper/2609.33078/) (2026)

## Abstract (arXiv)

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.