  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.
