Artificial neural network approximation of special functions: design, analysis and implementation

This work attempts the real-time hardware implementation of special functions used in fractional calculus (FC) like Gamma function, Mittag–Leffler function, Error and Complementary Error function, Gauss Hypergeometric function, and Dawson’s function using artificial neural network (ANN) on Field Programmable Gate Array (FPGA) platform. Special functions appear in the definition of fractional derivatives and integrals, and in the solution of fractional differential equations (FDEs). They are also a part of the output response of systems described with fractional dynamics. In spite of the rigorous development in the theory and numerical computations, there have not been much attempts toward the hardware implementation of special functions. The infinite summation definition of these functions and the complexity of their numerical algorithms make them difficult to implement in real-time on a limited memory hardware system. This work uses the ANN approximations of the special functions to carry out this task. ANNs with their interpolation and learning capability are ideal for modeling the special functions. It is shown with rigorous simulation and hardware exercise that the proposed novel methodology of hardware implementation of special functions using their ANN representations is very easy to execute. It is also shown that ANN models utilize much lower FPGA resources with reduced power consumption as compared to their conventional mode of implementation. The efficacy of the proposed procedure is demonstrated by implementing ANN approximation of special functions in various applications like implementation of fractional derivative and solution of FDEs.