Neural network as a function approximator and its application in solving differential equations

doi:10.1007/s10483-019-2429-8

Abstract

Abstract: A neural network (NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations (ODEs) and partial differential equations (PDEs) combined with the automatic differentiation (AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation (i.e., the Laplace equation).

Key words: stress structure, singularity, Z₁ region, stress triaxiality Rσ, CTOD, shear lip, interpolative method, degree parameter of the plane strain state, partial differential equation (PDE) solver, neural network (NN), function approximation, ordinary differential equation (ODE) solver

2010 MSC Number:

65D15

Zeyu LIU, Yantao YANG, Qingdong CAI. Neural network as a function approximator and its application in solving differential equations. Applied Mathematics and Mechanics (English Edition), 2019, 40(2): 237-248.

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