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

General partial differential equations, which can describe any complex functions, may be solved by means of the dimensional similarity analysis to model polynomial data relations of discrete data observations. Designed 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 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 allow. The sigmoidal function, commonly employed as an activation function in artificial neurons, may improve the polynomial and substituting derivative term abilities to approximate complicated periodic multi-variable or time-series functions in a system model.