We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains.

翻译：本文提出了一种基于深度学习的框架，用于在偏微分方程数值近似中弱实施边界条件。该方法建立在现有物理信息神经网络与深度Ritz方法的基础上，我们提出了Deep Uzawa算法，通过引入拉格朗日乘子来有效处理边界条件。此改进仅需微小的计算调整，却能确保增强的收敛特性，并可证明即使在奇异摄动问题中也能准确实施边界条件。我们提供了完整的数学分析以证明该格式的收敛性，并通过数值实验验证了Deep Uzawa算法的有效性，包括高维奇异摄动问题及定义在非凸区域上的问题。