The numerical solution of linear ordinary differential equations by feedforward neural networks

Abstract

It is demonstrated, through theory and examples, how it is possible to construct directly and noniteratively a feedforward neural network to approximate arbitrary linear ordinary differential equations. The method, using the hard limit transfer function, is linear in storage and processing time, and the L 2 norm of the network approximation error decreases quadratically 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 hard limit transfer function. However, the noniterative approach should also be applicable to the use of hyperbolic tangents, sigmoids, and radial basis functions.