Power series neural network solution for ordinary differential equations with initial conditions

Differential equations are very common in most academic fields. Modern digital control systems require fast on line and sometimes time varying solution schemes for differential equations. This paper presents new nonlinear adaptive numeric solutions for ordinary differential equations (ODE) with initial conditions. The main feature is to implement nonlinear polynomial expansions in a neural network-like adaptive framework. The transfer functions of the employed neural network follow a power series. The proposed technique does not use sigmoid or tanch non-linear transfer functions commonly adopted in conventional neural networks at the output. Instead, linear transfer functions are employed which leads to explicit power series formulae for the ODE solution. This allows extrapolation and interpolation which increase the dynamic numeric range for the solutions. The improved and accurate solutions for the proposed power series neural network (PSNN) are illustrated through simulated examples. It is shown that the performance of the proposed PSNN ODE solution outperforms existing conventional methods.