title: Constructing general partial differential equations using polynomial and neural networks.
abstract: Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.
authors.affiliation: VŠB-Technical University of Ostrava, Faculty of Electrical Engineering and Computer Science, Department of Computer Science, 17. listopadu 15/2172 Ostrava, Czech Republic. Electronic address: lzjavka@gmail.com.
authors.forename: Ladislav
authors.initials: L
authors.lastname: Zjavka
authors.affiliation: Department of Electrical & Computer Engineering, University of Alberta, Edmonton T6R 2V4 AB, Canada; Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia; Systems Research Institute, Polish Academy of Sciences Warsaw, Poland. Electronic address: wpedrycz@ualberta.ca.
authors.forename: Witold
authors.initials: W
authors.lastname: Pedrycz
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D000069550:Machine Learning; D008433:Mathematics; D016571:Neural Networks, Computer; D017711:Nonlinear Dynamics; D014887:Weather
keywords: Differential polynomial neural network; General partial differential equation composition; Multi-variable function approximation; Sum derivative term substitution