  Solving high-dimensional partial differential equations (PDEs) is a major
challenge in scientific computing. We develop a new numerical method for
solving elliptic-type PDEs by adapting the Q-learning algorithm in
reinforcement learning. Our "Q-PDE" algorithm is mesh-free and therefore has
the potential to overcome the curse of dimensionality. Using a neural tangent
kernel (NTK) approach, we prove that the neural network approximator for the
PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of
an infinite-dimensional ordinary differential equation (ODE) as the number of
hidden units $\rightarrow \infty$. For monotone PDE (i.e. those given by
monotone operators, which may be nonlinear), despite the lack of a spectral gap
in the NTK, we then prove that the limit neural network, which satisfies the
infinite-dimensional ODE, converges in $L^2$ to the PDE solution as the
training time $\rightarrow \infty$. More generally, we can prove that any fixed
point of the wide-network limit for the Q-PDE algorithm is a solution of the
PDE (not necessarily under the monotone condition). The numerical performance
of the Q-PDE algorithm is studied for several elliptic PDEs.
