The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by parameterizing a set of functions. Various approaches, such as the deep Ritz method and physics-informed neural networks, have been developed for numerical solutions. Training algorithms, including gradient descent and greedy algorithms, have been proposed to solve the resulting optimization problems. In this paper, we focus on the variational formulation of the problem and propose a Gauss- Newton method for computing the numerical solution. We provide a comprehensive analysis of the superlinear convergence properties of this method, along with a discussion on semi-regular zeros of the vanishing gradient. Numerical examples are presented to demonstrate the efficiency of the proposed Gauss-Newton method.

翻译：基于机器学习方法的微分方程数值求解已获得广泛关注。通过参数化一组函数，神经网络离散化已成为求解微分方程的有力工具。诸如深度Ritz方法与物理信息神经网络等多种方法已被开发用于数值求解。梯度下降法与贪心算法等训练算法也被提出以求解由此产生的优化问题。本文聚焦于问题的变分形式，提出了一种用于计算数值解的高斯-牛顿方法。我们对该方法的超线性收敛性质进行了全面分析，并讨论了消失梯度的半正则零点。数值算例展示了所提高斯-牛顿方法的有效性。