**
Solution of Nonlinear 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, nonlinear
differential equations, basis functions.

**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.

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