Andrew J. Meade, Jr.,

Submitted to

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

**Abstract**: A method is developed for manually
constructing recurrent artificial neural networks to model the fusion of
experimental data and mathematical models of physical systems. The construction
requires the use of Generalized Tikhonov Regularization (GTR) and 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.
GTR provides a rational means of combining theoretical models, computational
data, and experimental measurements into a global representation of a domain.
Attention is focused on a second-order nonlinear ordinary differential
equation, which governs the classic 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. GTR is then used to smoothly merge the response
of the RANN and experimental data. 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 was supported under Office of Naval Research grant N00014-95-1-0741.