A substitution of the general partial differential equation with extended polynomial networks

Abstract

General partial differential equations, which can describe any complex functions, may be solved by an adapted method of the similarity analysis that models polynomial data relations of discrete observations. The proposed new differential polynomial networks define and substitute for a selective form of the general partial differential equation using fraction derivative units to model an unknown system or pattern. Convergent series of relative derivative substitution terms, produced in all network layers, describe the partial derivative changes of some combinations of input variables to generalize elementary polynomial data relations. The general differential equation is decomposed into polynomial network backward structure, which defines simple and composite sum derivative terms in respect of previous layers variables. The proposed method enables to form more complex and varied derivative selective series models than standard soft-computing techniques. The sigmoidal function, commonly employed as an activation function in artificial neurons, may improve the abilities of the polynomial networks and substituting derivative terms to approximate complicated periodic multi-variable or time-series functions in a system model.