**
The Numerical Solution of Linear Ordinary
Differential Equations by Feedforward Neural Networks
**

Andrew J. Meade, Jr. and Alvaro A. Fernandez ,

Submitted to *Journal of Mathematical and Computer Modelling*, 1993.

**Keywords:**:
artificial neural networks, neural computation, differential equations, basis functions.

**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.

This work was supported under NASA grant NAG 1-1433.