Before revision:  Our proof is based on constructing a neural network which simulates gradient descent  in an appropriate Hilbert space which converges to the solution of the PDE.
After revision:  Our proof is based on constructing a neural network which simulates gradient descent in an appropriate Hilbert space ) by growing a neural network architecture whose iterates each participate as sub-networks in their (slightly larger) successors, and converge to the solution of the PDE.