Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we specifically focus on employing a three-layer tanh neural network within the framework of the deep Ritz method(DRM) to solve second-order elliptic equations with three different types of boundary conditions. We perform projected gradient descent(PDG) to train the three-layer network and we establish its global convergence. To the best of our knowledge, we are the first to provide a comprehensive error analysis of using overparameterized networks to solve PDE problems, as our analysis simultaneously includes estimates for approximation error, generalization error, and optimization error. We present error bound in terms of the sample size $n$ and our work provides guidance on how to set the network depth, width, step size, and number of iterations for the projected gradient descent algorithm. Importantly, our assumptions in this work are classical and we do not require any additional assumptions on the solution of the equation. This ensures the broad applicability and generality of our results.

翻译：机器学习是一个快速发展的领域，在各个领域都有广泛的应用。其中一个重要的研究方向是利用深度学习技术求解偏微分方程。本文重点研究在深度Ritz方法框架下，采用三层tanh神经网络求解具有三种不同边界条件的二阶椭圆方程。我们采用投影梯度下降训练该三层网络，并建立了其全局收敛性。据我们所知，这是首个对使用过参数化网络求解偏微分方程问题进行完整误差分析的研究，其分析同时包含了逼近误差、泛化误差和优化误差的估计。我们给出了关于样本量$n$的误差界，并为投影梯度下降算法的网络深度、宽度、步长和迭代次数设置提供了指导。重要的是，本文的假设均为经典假设，不需要对方程的解附加任何额外假设，这确保了结果的广泛适用性和一般性。