Complex System Modeling with General Differential Equations Solved by Means of Polynomial Networks

Abstract

Differential equations can describe physical and natural systems, which behavior only explicit exact functions are not able to model. Complex dynamic systems are characterized by a high variability of time-fluctuating data relations of a great number of state variables. Systems of differential equations can describe them but they are too unstable to be modeled unambiguously by means of standard soft computing techniques. In some cases the correct form of a differential equation might absent or it is difficult to express. Differential polynomial neural network is a new neural network type, which forms and solves an unknown general partial differential equation of an approximation of a searched function, described by discrete data observations. It generates convergent sum series of relative partial polynomial derivative terms, which can substitute for a partial or/and ordinary differential equation solution. This type of non-linear regression decomposes a system model, described by the general differential equation, into low order composite partial polynomial fractions in an additive series solution. The differential network can model the dynamics of the complex weather system, using only several input variables in some cases. Comparisons were done with the recurrent neural network, often applied for simple and solid time-series models.