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受到广泛关注。通过参数化函数族实现的神经网络离散化，已成为求解微分方程的有力工具。目前已发展出深度Ritz方法和物理信息神经网络等多种数值求解方法，并提出了梯度下降法和贪心算法等训练算法来求解相应的优化问题。本文聚焦于问题的变分公式，提出了一种用于计算数值解的高斯-牛顿方法。我们全面分析了该方法的超线性收敛特性，并对消失梯度半正则零点进行了讨论。数值算例验证了所提高斯-牛顿方法的有效性。