Applied Mathematics and Mechanics

, Volume 40, Issue 2, pp 237–248 | Cite as

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

  • Zeyu Liu
  • Yantao Yang
  • Qingdong CaiEmail author


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

neural network (NN) function approximation ordinary differential equation (ODE) solver partial differential equation (PDE) solver 

Chinese Library Classification


2010 Mathematics Subject Classification



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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory for Turbulence and Complex SystemPeking UniversityBeijingChina
  2. 2.Department of Mechanics and Engineering Science, College of EngineeringPeking UniversityBeijingChina
  3. 3.Center for Applied Physics and Technology, College of EngineeringPeking UniversityBeijingChina

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