Neural Q-Learning for Solving PDEs

Abstract

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. To solve PDEs with Dirichlet boundary condition, 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 PDEs (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, strongly 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.