# An augmented Lagrangian deep learning method for variational problems with essential boundary conditions

arXiv: https://arxiv.org/abs/2106.14348 · HTML: https://mlsanity.com/paper/2106.14348/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2021.

## Built on by (most cited first, 11 of 11)

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- [WANCO: Weak Adversarial Networks for Constrained Optimization problems](https://mlsanity.com/paper/2407.03647/) (2024)
- [MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling](https://mlsanity.com/paper/2212.12984/) (2022)
- [LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process](https://mlsanity.com/paper/2507.22493/) (2025)
- [Learning solutions to some toy constrained optimization problems in infinite dimensional Hilbert spaces](https://mlsanity.com/paper/2401.01306/) (2024)

## Abstract (arXiv)

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.