Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods, for example, offer a straightforward implementation but introduces additional complexity due to the need for hyper-parameter tuning; moreover, the use of a large penalty parameter can lead to artificial extra stiffness, complicating the optimization process. In this paper, we propose a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures. We demonstrate the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques. Numerical results are provided to substantiate the efficiency and robustness of the proposed method.

翻译：深度神经网络方法在求解偏微分方程方面展现出潜力。然而，与传统数值方法不同，它们在施加本质边界条件方面面临挑战。广泛采用的惩罚类方法虽然实现简单，但由于需要超参数调优而引入了额外的复杂性；此外，使用大惩罚参数可能导致人为的额外刚度，使优化过程复杂化。本文提出了一种新颖的、本质性的方法，通过受本质结构启发的框架来施加本质边界条件。我们以应用于泊松问题的深度Ritz方法为例，展示了该方法的有效性，并指出其可推广至更一般的方程及其他深度学习技术。数值结果验证了所提方法的效率与鲁棒性。