Neural Q-learning for solving PDEs

• URL: http://arxiv.org/abs/2203.17128v2
• Date: Sun, 25 Jun 2023 03:32:33 GMT
• Title: Neural Q-learning for solving PDEs
• Authors: Samuel N. Cohen and Deqing Jiang and Justin Sirignano
• Abstract summary: 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. The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. 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.

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