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.

翻译：求解高维偏微分方程（PDEs）是科学计算中的一项重大挑战。我们通过将强化学习中的Q-learning算法应用于椭圆型PDE求解，发展了一种新的数值方法。我们的"Q-PDE"算法无需网格，因此有望克服维数灾难。利用神经正切核（NTK）方法，我们证明了使用Q-PDE算法训练的PDE解神经网络逼近器在隐单元数→∞时会收敛到无穷维常微分方程（ODE）的轨迹。对于单调PDE（即由单调算子给出的PDE，可能为非线性），尽管NTK缺乏谱间隙，我们进一步证明了满足该无穷维ODE的极限神经网络在训练时间→∞时会在L²范数下收敛到PDE解。更一般地，我们能够证明Q-PDE算法宽网络极限的任何不动点均为PDE的解（不必然需要单调条件）。最后，我们通过多个椭圆型PDE案例研究了Q-PDE算法的数值性能。