The neural network collocation method for solving partial differential equations

This paper presents a meshfree collocation method that uses deep learning to determine the basis functions as well as their corresponding weights. This method is shown to be able to approximate elliptic, parabolic, and hyperbolic partial differential equations for both forced and unforced systems, as well as linear and nonlinear partial differential equations. By training a homogeneous network and particular network separately, new forcing functions are able to be approximated quickly without the burden of retraining the full network. The network is demonstrated on several numerical examples including a nonlinear elasticity problem. In addition to providing meshfree approximations to strong form partial differential equations directly, this technique could also provide a foundation for deep learning methods to be used as preconditioners to traditional methods, where the deep learning method will get close to a solution and traditional solvers can finish the solution.