Design of stochastic solvers based on genetic algorithms for solving nonlinear equations

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

“…The above results mean L(t) can directly converge to zero at most after a time period Thus, we can conclude that neural state x(t) of FTCND model (7), starting from randomly-generated initial state x(0), can converge to the time-varying theoretical solution of nonlinear equation (1) in finite time t f . This completes the proof.…”

“…In addition, taking advantage of the nonlinearity, a properly-designed nonlinear activation function often outperforms the linear one in convergence rate. However, this OZND model (4) with the suggested activation functions cannot converge to the time-varying theoretical solution of time-varying nonlinear equation (1) in finite time, which may limit its applications in real-time calculation. Therefore, in this section, we aim at developing a specially-constructed nonlinear activation function, which can endow OZND model (4) with a finite-time convergence for solving timevarying nonlinear equation (1).…”

“…Therefore, in this section, we aim at developing a specially-constructed nonlinear activation function, which can endow OZND model (4) with a finite-time convergence for solving timevarying nonlinear equation (1). Inspired by the study on finite-time control of autonomous systems and finite-time convergence of recurrent neural dynamics [22,23,24,25,26,27], we present a weighted sign-bi-power activation function [28,29] to accelerate OZND model (4) to finite-time convergence to the time-varying theoretical solution of nonlinear equation (1). The weighted sign-bi-power activation function wspb(·) is defined as follows [28,29]:…”

“…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].…”

“…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].…”