A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks
This addresses numerical PDE solving for computational science, but it is incremental as it builds on existing neural network and probabilistic approaches.
The authors tackled solving elliptic partial differential equations by introducing a deep neural network method that uses a probabilistic representation and reinforcement learning, achieving results demonstrated on test problems like a corner singularity and chemotaxis model.
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, an interface problem, and an application to a chemotaxis population model.