Solution of nonlinear ordinary differential equations by feedforward neural networks

Authors:
A. J. Meade, Jr.;A. A. Fernandez
Affiliations:
Department of Mechanical Engineering and Materials Science Rice University, Houston, TX, 77251-1892, U.S.A.;Department of Mechanical Engineering and Materials Science Rice University, Houston, TX, 77251-1892, U.S.A.
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
1994

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Abstract

It is demonstrated, through theory and numerical examples, how it is possible to directly construct a feedforward neural network to approximate nonlinear ordinary differential equations without the need for training. The method, utilizing a piecewise linear map as the activation function, is linear in storage, and the L"2 norm of the network approximation error decreases monotonically with the increasing number of hidden layer neurons. The construction requires imposing certain constraints on the values of the input, bias, and output weights, and the attribution of certain roles to each of these parameters. All results presented used the piecewise linear activation function. However, the presented approach should also be applicable to the use of hyperbolic tangents, sigmoids, and radial basis functions.