WITHDRAWN Functional Differential Equation Model for Dynamic System

Music City Center 

Ordinary Differential Equations (ODEs) are commonly used in modeling dynamic systems. However, one major limitation of the ODE model is that it assumes the derivatives of the system only depend on the concurrent values. This concurrent assumption may oversimplify the mechanisms of dynamic systems and limit the applicability of differential equations. To address this, we propose a general Functional Differential Equation (FDE) model which allows the derivative to explicitly depend on both the current value and a historical segment of the system through an unknown operator which maps historical curves to scalars. To estimate the FDE model from noisy observations, we propose the Functional Neural Networks (FNNs) with a smooth hidden layer and establish their universal approximate property: the FNNs can universally approximate the operator in FDE and the solution to the approximate FDE can be uniformly and arbitrarily close to the solution to the original FDE. We propose a new method based on the changes of the dynamic system on moving windows to construct the FNN, and then make forecasts by solving the approximate FDE.

differential equation

dynamic systems

functional differential equation

functional universal approximation theorem

functional neural networks 

Section on Statistical Learning and Data Science