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Neural network as a function approximator and its application in solving differential equations 被引量:3

Neural network as a function approximator and its application in solving differential equations
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摘要 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). 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).
出处 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2019年第2期237-248,共12页 应用数学和力学(英文版)
基金 Project supported by the National Natural Science Foundation of China(No.11521091)
关键词 neural network(NN) FUNCTION approximation ordinary DIFFERENTIAL equation(ODE)solver partial DIFFERENTIAL equation(PDE)solver neural network(NN) function approximation ordinary differential equation(ODE) solver partial differential equation(PDE) solver
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