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The numerical solution of differential equations using machine learning-basedapproaches has gained significant popularity. Neural network-baseddiscretization has emerged as a powerful tool for solving differentialequations by parameterizing a set of functions. Various approaches, such as thedeep Ritz method and physics-informed neural networks, have been developed fornumerical solutions. Training algorithms, including gradient descent and greedyalgorithms, have been proposed to solve the resulting optimization problems. Inthis paper, we focus on the variational formulation of the problem and proposea Gauss- Newton method for computing the numerical solution. We provide acomprehensive analysis of the superlinear convergence properties of thismethod, along with a discussion on semi-regular zeros of the vanishinggradient. Numerical examples are presented to demonstrate the efficiency of theproposed Gauss-Newton method.

Gauss Newton method for solving variational problems of PDEs with neural network discretizaitons