**
A Recurrent Artificial Neural Network Model of Duffing's Equation without Training
**

Andrew J. Meade, Jr. and Rafael Moreno,

Submitted to *International Journal of Smart Engineering System Design*, 1996.

**Keywords:**:
recurrent artificial neural networks, neural
computation, differential equations, chaos, network training.

**Abstract**:
A method is developed for constructing recurrent artificial neural
networks to model physical systems.
The construction requires imposing certain constraints on the values of the
input, bias, and output weights. The attribution of certain roles to each of these
parameters allows for mapping a polynomial approximation into an artificial
neural network architecture. Attention is focused on a second-order nonlinear
ordinary differential equation, which governs the well known Duffing's oscillator. The nonlinear ordinary differential equation is
modelled by the recurrent artificial neural network architecture in conjunction
with the popular hyperbolic tangent transfer function. Moreover, this approach
is shown to be capable of incorporating other smooth neuron transfer functions,
as long as they can be described by a Taylor series expansion.
Numerical examples are presented illustrating the accuracy and utility of the
method.

This work is supported under NASA grant NAG 9-719 and ONR grant
N00014-95-1-0741.