The neural network collocation method for solving partial differential equations
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ORIGINAL ARTICLE
The neural network collocation method for solving partial differential equations Adam R. Brink1
•
David A. Najera-Flores2 • Cari Martinez3
Received: 19 March 2020 / Accepted: 2 September 2020 Ó This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2020
Abstract 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. Keywords PDE Collocation Meshfree Basis function
1 Introduction Partial differential equations (PDEs) have been used by engineers and scientists to describe the world around us since first written by Isaac Newton and Gottfried Leibniz in the early 1670s [1]. While some simple PDEs can be solved analytically, most researchers currently rely on numerical methods. Many numerical methods exist for solution, with
& Adam R. Brink [email protected] David A. Najera-Flores [email protected] Cari Martinez [email protected] 1
Department of Structural Mechanics, Sandia National Laboratories, P.O. Box 5800, MS 0346, Albuquerque, NM 57185-0346, USA
2
Department of Component Sciences and Mechanics, ATA Engineering, Inc, 13290 Evening Creek Drive S, San Diego, CA 92128, USA
3
Department of Applied Machine Learning, Sandia National Laboratories, P.O. Box 5800, MS 0346, Albuquerque, NM 57185-0346, USA
the most common being the finite difference method (FDM) [2], finite element method (FEM) [3, 4], and finite volume methods (FVMs) [5]. The ‘finite’ designation in these methods alludes to their major shortcoming: Domains must be first be discretized, also called ‘meshed,’ for these methods to produce a solution. For simplistic domains, such as squares or cubes, the task of discretization is a trivial matter. But when encountering a complex real-world structure, such as a jet airplane, it takes experienced analysts months to produce a high-quality mesh. A solution to the problem of discretization was first proposed by [6] and [7], via smoothed particle hydrodynamics (SPH). In this ‘meshfree’ method, the solution is no longer dependent upon traditional element connectivi
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