  This paper analyzes the convergence rate of a deep Galerkin method for the
weak solution (DGMW) of second-order elliptic partial differential equations on
$\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions,
respectively. In DGMW, a deep neural network is applied to parametrize the PDE
solution, and a second neural network is adopted to parametrize the test
function in the traditional Galerkin formulation. By properly choosing the
depth and width of these two networks in terms of the number of training
samples $n$, it is shown that the convergence rate of DGMW is
$\mathcal{O}(n^{-1/d})$, which is the first convergence result for weak
solutions. The main idea of the proof is to divide the error of the DGMW into
an approximation error and a statistical error. We derive an upper bound on the
approximation error in the $H^{1}$ norm and bound the statistical error via
Rademacher complexity.
