Finding all roots of 2 × 2 nonlinear algebraic systems using back-propagation neural networks

A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation

“…Specifically, Fig. 1 shows the transient behavior of neural states x(t) of GND model (2) and OZND model (4) during the solution process of time-varying nonlinear equation (9). As seen from Fig.…”

“…As seen from Fig. 1(a), starting from 20 randomly-generated initial states within [−5, 5], neural states x(t) of GND model (2) do not fit well with any one of the time-varying theoretical solutions x * 1 (t) and x * 2 (t). In contrast, under the same conditions, neural states x(t) of OZND model (4) can converge to one of the time-varying theoretical solutions [i.e., x * 1 (t) or x * 2 (t)], which is shown evidently in Fig.…”

“…Many phenomena in physics and other fields are often described by nonlinear equations in the form of f (x) = 0 [1,2]. When we want to understand the physical mechanism of phenomena in nature, exact solutions for the nonlinear equations have to be explored [3,4,5,6].…”

“…When we want to understand the physical mechanism of phenomena in nature, exact solutions for the nonlinear equations have to be explored [3,4,5,6]. Thus, considerable attention has been focused to derive exact solutions of nonlinear equations both analytically and numerically [1,2,7,8,9]. For example, several iterative numerical algorithms have been developed using quite different techniques such as quadrature formulas, homotopy, Taylor's series, interpolation, decomposition and its various modification [7,8,9].…”

Short-term wind power prediction based on Hybrid Neural Network and chaotic shark smell optimization

“…Thus, to apply the chaotic operator for R3, the following equation is used in addition to Eq. (16): (17) In other words, is used as R3. Considering the range of indices in Eq.…”