Zhang Dynamics Models Research Articles

From Davidenko Method to Zhang Dynamics for Nonlinear Equation Systems Solving

The solving of nonlinear equation systems (e.g., complex transcendental dispersion equation systems in waveguide systems) is a fundamental topic in science and engineering. Davidenko method has been used by electromagnetism researchers to solve time-invariant nonlinear equation systems (e.g., the aforementioned transcendental dispersion equation systems). Meanwhile, Zhang dynamics (ZD), which is a special class of neural dynamics, has been substantiated as an effective and accurate method for solving nonlinear equation systems, particularly time-varying nonlinear equation systems. In this paper, Davidenko method is compared with ZD in terms of efficiency and accuracy in solving time-invariant and time-varying nonlinear equation systems. Results reveal that ZD is a more competent approach than Davidenko method. Moreover, discrete-time ZD models, corresponding block diagrams, and circuit schematics are presented to facilitate the convenient implementation of ZD by researchers and engineers for solving time-invariant and time-varying nonlinear equation systems online. The theoretical analysis and results on Davidenko method, ZD, and discrete-time ZD models are also discussed in relation to solving time-varying nonlinear equation systems.

A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients

The recently-proposed Zhang dynamics (ZD) has been proven to achieve success for solving the linear-equality constrained time-varying quadratic program ideally when time goes to infinity. The convergence performance is a significant improvement, as compared to the gradient-based dynamics (GD) that cannot make the error converge to zero even after infinitely long time. However, this ZD model with the suggested activation functions cannot reach the theoretical time-varying solution in finite time, which may limit its applications in real-time calculation. Therefore, a nonlinearly-activated neurodynamic model is proposed and studied in this paper for real-time solution of the equality-constrained quadratic optimization with nonstationary coefficients. Compared with existing neurodynamic models (specifically the GD model and the ZD model) for optimization, the proposed neurodynamic model possesses the much superior convergence performance (i.e., finite-time convergence). Furthermore, the upper bound of the finite convergence time is derived analytically according to Lyapunov theory. Both theoretical and simulative results verify the efficacy and superior of the nonlinearly-activated neurodynamic model, as compared to these of the GD and ZD models.

Solving time-varying nonlinear inequalities using continuous and discrete-time Zhang dynamics

A special kind of neural dynamics, termed Zhang dynamics (ZD), is proposed, generalized and investigated for the online solution of time-varying scalar-valued nonlinear inequalities by following Zhang et al.’s design method. The continuous-time ZD (CTZD) model based on an exponent-type design formula can be guaranteed to exponentially converge to the time-varying solution set of the problem in an error-free manner. For potential hardware implementation on digital circuits, the corresponding discrete-time ZD (DTZD) model is generated through the well-known Euler forward difference rule. Newton-type algorithm is also developed for comparison purposes. In addition, the simplified CTZD (S-CTZD) and DTZD (S-DTZD) models are developed for solving static scalar-valued nonlinear inequalities. Numerical simulative examples further demonstrate and verify the efficacy of the ZD models for solving time-varying and static scalar-valued nonlinear inequalities. Besides, the DTZD model possesses the lower complexity and higher accuracy, as compared with the Newton-type algorithm.

Discrete-time ZD, GD and NI for solving nonlinear time-varying equations

A special class of neural dynamics called Zhang dynamics (ZD), which is different from gradient dynamics (GD), has recently been proposed, generalized, and investigated for solving time-varying problems by following Zhang et al.'s design method. In view of potential digital hardware implemetation, discrete-time ZD (DTZD) models are proposed and investigated in this paper for solving nonlinear time-varying equations in the form of $f(x,t)=0$ . For comparative purposes, the discrete-time GD (DTGD) model and Newton iteration (NI) are also presented for solving such nonlinear time-varying equations. Numerical examples and results demonstrate the efficacy and superiority of the proposed DTZD models for solving nonlinear time-varying equations, as compared with the DTGD model and NI.

Solving for time-varying and static cube roots in real and complex domains via discrete-time ZD models

Different from conventional gradient-based neural dynamics, a special type of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static (or termed, time-invariant) problems. The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error function, instead of the elimination of a square-based positive (or at least lower-bounded) energy function usually associated with gradient dynamics (GD). In this paper, we generalize, propose and investigate the continuous-time ZD model and its discrete-time models in two situations (i.e., the time-derivative of the coefficient being known or unknown) for time-varying cube root finding, including the complex-valued continuous-time ZD model for finding cube roots in complex domain. In addition, to find the static scalar-valued cube root, a simplified continuous-time ZD model and its discrete-time model are generated. By focusing on such a static problem solving, Newton-Raphson iteration is found to be a special case of the discrete-time ZD model by utilizing the linear activation function and fixing the step-size value to be 1. Computer-simulation and testing results demonstrate the efficacy of the proposed ZD models (including real-valued ZD models and complex-valued ZD models) for time-varying and static cube root finding, as well as the link and new explanation to Newton-Raphson iteration.

Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving

Along with neural dynamics (based on analog solvers) widely arising in scientific computation and optimization fields in recent decades which attracts extensive interest and investigation of researchers, a novel type of neural dynamics, called Zhang dynamics (ZD), has been formally proposed by Zhang et al. for the online solution of time-varying problems. By following Zhang et al.’s neural-dynamics design method, the ZD model, which is based on an indefinite Zhang function (ZF), can guarantee the exponential convergence performance for the online time-varying problems solving. In this paper, different indefinite Zhang functions, which can lead to different ZD models, are proposed and developed as the error-monitoring functions for the time-varying reciprocal problem solving. Additionally, for the goal of developing the floating-point processors or coprocessors for the future generation of computers, the MATLAB Simulink modeling and simulative verifications of such different ZD models are further presented for online time-varying reciprocal solving. The modeling results substantiate the efficacy of such different ZD models for time-varying reciprocal solving.

Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former's link and new explanation to Newton–Raphson iteration

Different from conventional gradient-based neural dynamics, a special class of neural dynamics have been proposed by Zhang et al. since 12 March 2001 for online solution of time-varying and static (or termed, time-invariant) problems (e.g., nonlinear equations). The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error-function, instead of the elimination of a square-based positive or at least lower-bounded energy-function usually associated with gradient dynamics (GD) and/or Hopfield-type neural networks. In this paper, we generalize, develop, investigate and compare the continuous-time ZD (CTZD) and GD models for online solution of time-varying and static square roots. In addition, a simplified continuous-time ZD (S-CTZD) and discrete-time ZD (DTZD) models are generated for static scalar-valued square roots finding. In terms of such scalar square roots finding problem, the Newton iteration (also termed, Newton–Raphson iteration) is found to be a special case of the DTZD models (by focusing on the static-problem solving, utilizing the linear activation function and fixing the step-size to be 1). Computer-simulation results via a power-sigmoid activation function further demonstrate the efficacy of the ZD solvers for online scalar (time-varying and static) square roots finding, in addition to the DTZD's link and new explanation to Newton–Raphson iteration.

Continuous and discrete time Zhang dynamics for time-varying 4th root finding

Recently, a special class of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static nonlinear equations. Different from eliminating a square-based positive error-function associated with gradient-based dynamics (GD), the design method of Zhang dynamics (ZD) is based on the elimination of an indefinite (unbounded) error-function. In this paper, for the purpose of online solution of time-varying 4th root, both continuous-time ZD (CTZD) and discrete-time ZD (DTZD) models are developed and investigated. In addition, power-sigmoid activation function is exploited in Zhang dynamics, which makes ZD models possess the property of superior convergence and better accuracy. To summarize generalization for possible widespread application, such approach is further extended to general time-varying nonlinear equations solving. Computer-simulation results demonstrate the efficacy of the ZD models for finding online time-varying 4th root and solving general time-varying equations.