论文标题
用于解决PDE的神经Q学习
Neural Q-learning for solving PDEs
论文作者
论文摘要
解决高维偏微分方程（PDE）是科学计算中的主要挑战。我们开发了一种新的数值方法，用于通过在增强学习中调整Q学习算法来求解椭圆型PDE。我们的“ Q-PDE”算法是无网格的，因此有可能克服维度的诅咒。使用神经切线核（NTK）方法，我们证明了使用Q-PDE算法训练的PDE溶液的神经网络近似器会收敛到无限二维的常见差分方程（ODE）的轨迹，作为隐藏单位$ \ rightarrow $ \ rightarrow \ iffty $。对于单调PDE（即单调操作员给出的那些可能是非线性的），尽管NTK缺乏光谱差距，然后我们证明，满足InfInite-二维驱动器的极限神经网络在$ l^2 $中的收敛到PDE解决方案，作为培训时间$ \ rightarrow $ \ rightarrow $ \ firfty $。更普遍地，我们可以证明Q-PDE算法的宽网络极限的任何固定点是PDE的解决方案（不一定在单调条件下）。研究了几种椭圆PDE的Q-PDE算法的数值性能。
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.
