In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms on some toy problems originating in either calculus of variations or physics. We demonstrate that both methods are able to produce decent approximations for the test problems and are comparable in terms of different errors. Leveraging the common occurrence of the Lagrange multiplier update rule being computationally less expensive than solving subproblems in the penalty method, we achieve significant speedups in cases when the output of the constraint function is itself a function.

翻译：本文展示了两种流行理论约束优化算法在无限维希尔伯特空间中的深度学习实现，即惩罚法和增广拉格朗日法。我们在源于变分法或物理学的若干玩具问题上测试了这些算法。实验表明，两种方法均能为测试问题生成合理的近似解，且在各类误差指标上具有可比性。利用拉格朗日乘子更新规则计算成本通常低于惩罚法子问题求解这一常见特性，我们实现了在约束函数输出本身为函数时的显著加速。