Mingzhi Mao

Zeroing Dynamics, Gradient Dynamics, and Newton Iterations

Neural networks and neural dynamics are powerful approaches for the online solution of mathematical problems arising in many areas of science, engineering, and business. Compared with conventional gradient neural networks that only deal with static problems of constant coefficient matrices and vectors, the authors new method called zeroing dynamics solves time-varying problems. Zeroing Dynamics, Gradient Dynamics, and Newton Iterations is the first book that shows how to accurately and efficiently solve time-varying problems in real-time or online using continuous- or discrete-time zeroing dynamics. The book brings together research in the developing fields of neural networks, neural dynamics, computer mathematics, numerical algorithms, time-varying computation and optimization, simulation and modeling, analog and digital hardware, and fractals. The authors provide a comprehensive treatment of the theory of both static and dynamic neural networks. Readers will discover how novel theoretical results have been successfully applied to many practical problems. The authors develop, analyze, model, simulate, and compare zeroing dynamics models for the online solution of numerous time-varying problems, such as root finding, nonlinear equation solving, matrix inversion, matrix square root finding, quadratic optimization, and inequality solving.

Common nature of learning between BP-type and Hopfield-type neural networks

Being two famous neural networks, the error back-propagation (BP) algorithm based neural networks (i.e., BP-type neural networks, BPNNs) and Hopfield-type neural networks (HNNs) have been proposed, developed, and investigated extensively for scientific research and engineering applications. They are different from each other in a great deal, in terms of network architecture, physical meaning and training pattern. In this paper of literature-review type, we present in a relatively complete and creative manner the common natures of learning between BP-type and Hopfield-type neural networks for solving various (mathematical) problems. Specifically, comparing the BPNN with the HNN for the same problem-solving task, e.g., matrix inversion as well as function approximation, we show that the BPNN weight-updating formula and the HNN state-transition equation turn out to be essentially the same. Such interesting phenomena promise that, given a neural-network model for a specific problem solving, its potential dual neural-network model can thus be developed.

Z-type neural-dynamics for time-varying nonlinear optimization under a linear equality constraint with robot application

Abstract Nonlinear optimization is widely important for science and engineering. Most research in optimization has dealt with static nonlinear optimization while little has been done on time-varying nonlinear optimization problems. These are generally more complicated and demanding. We study time-varying nonlinear optimizations with time-varying linear equality constraints and adapt Z-type neural-dynamics (ZTND) for solving such problems. Using a Lagrange multipliers approach we construct a continuous ZTND model for such time-varying optimizations. A new four-instant finite difference (FIFD) formula is proposed that helps us discretize the continuous ZTND model with high accuracy. We propose the FDZTND-K and FDZTND-U discrete models and compare their quality and the advantage of the FIFD formula with two standard Euler-discretization ZTND models, called EDZTND-K and EDZTND-U that achieve lower accuracy. Theoretical convergence of our continuous and discrete models is proved and our methods are tested in numerical experiments. For a real world, we apply the FDZTND-U model to robot motion planning and show its feasibility in practice.

Combining WASP and ASF algorithms to forecast a Japan earthquake with Mj 7.2 or above

An earthquake-forecasting attempt is presented in this work via combining the weights and structure policy (WASP) and addition-subtraction frequency (ASF) algorithms. Specifically, based on the application of three-layer feedforward neuronets equipped with WASP algorithm, further using ASF algorithm, this work attempts to forecast a Japan earthquake with Mj 7.2 or above. Note that past earthquake dates are the only data used in this study. The feasibility and effectiveness of this attempt are verified via the numerical experiments with consistency analysis on obtained dates. Besides, according to the experimental results, an earthquake with Mj 7.2 or above may occur in August 2016 in Japan. Furthermore, with the highest possibility, the date of such an earthquake may occur is August 9, 2016.

USPD Doubling or Declining in Next Decade Estimated by WASD Neuronet Using Data as of October 2013

Recently, the total public debt outstanding (TPDO) of the United States has increased rapidly, and to more than \(\

New formula of 4-instant g-square finite difference (4IgSFD) applied to time-variant matrix inversion

17\) trillion on October 18, 2013. It is important and necessary to conduct the TPDO projection for better policies making and more effective measurements taken. In this paper, we present the ten-year projection for the public debt of the United States (termed also the US public debt, USPD) via a 3-layer feed-forward neuronet. Specifically, using the calendar year data on the USPD from the Department of the Treasury, the neuronet is trained, and then is applied to projection. Via a series of numerical tests, we find that there are several possibilities of the change of the USPD in the future, which are classified into two categories in terms of projection trend: the continuous-increase trend and the increase-peak-decline trend. In the most possible situation, the neuronet indicates that the TPDO of the United States is projected to increase, and it will double in 2019 and double again in 2024.

Cart Velocity Tracking of General IPC Model Using ZG Control Compared with Cart Path Tracking

As the time-variant matrix inversion is considered to be one of basic problems widely encountered in a variety of scientific and engineering fields, a new formula of 4-instant g-square finite difference (4IgSFD) for the time-variant matrix inversion is proposed and investigated, where g denotes the sampling gap. Note that the new formula is based on Taylor expansion instead of Lagrange interpolation, and has a truncation error of O(g2). According to this novel formula, a new discrete-time Zhang dynamics (DTZD) model termed 4IgSFD-type DTZD model is derived. The model uses the present and previous information to calculate the inverse of time-variant matrix for the next (or say, future) time instant with a high calculative precision. Then, the stability and convergence of the 4IgSFD-type DTZD model are guaranteed theoretically. Finally, we take different types of nonsingular time-variant matrices with different dimensions as testing examples and use different values of g in numerical experiments. The numerical experiment results show the efficacy of the 4IgSFD-type DTZD model for time-variant matrix inversion with truncation error being O(g3).

A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals

As a typical model, the inverted pendulum on a cart (IPC) has nonlinear, unstable and non-minimum-phase characteristics and is investigated widely for tracking control problem solving. In this paper, we investigate a more general IPC model, in which the link of the pendulum has nonzero mass. Due to device velocity constrained, it is important and meaningful to investigate and analyze the velocity tracking control. To achieve velocity tracking control purposes, the presented ZG controllers consist of controllers of z1g0 and z1g1 types, named by the numbers of times of using Zhang dynamics (ZD) and gradient dynamics (GD) methods. Computer simulations and numerical experiments substantiate the feasibility and effectiveness of the ZG controllers for the velocity tracking control of the general IPC model. Furthermore, the superiority [in conquering the singularity problem (i.e., Division-by-zero problem)] of z1g1 controller for velocity tracking control of the general IPC model is well substantiated via comparative simulation and numerical experiment results. At last, the ZG controllers for cart velocity tracking are compared with those for cart path tracking.

Complex ZNN and GNN models for time-varying complex quadratic programming subject to equality constraints

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

ZG control for nonlinear system 2-output tracking with GD used additionally once more

Zhang neural network (ZNN) has shown powerful abilities to solve a great variety of time-varying problems in the real domain. In this paper, to solve the time-varying complex quadratic programming (QP) problems in the complex domain, a new type of complex-valued ZNN is further developed and investigated. Specifically, by defining two different complex-valued error functions (termed Zhang functions), two complex ZNN models are proposed and investigated for solving the time-varying complex QP subject to complex-valued linear-equality constraints. It is theoretically proved that such two complex ZNN models globally and exponentially converge to the time-varying theoretical optimal solution of the time-varying complex QP. For comparison, the conventional gradient neural network (GNN) is developed from the real to the complex domains and then is exploited for solving the time-varying complex QP problems. Computational simulation results verify the efficacy of complex ZNN models for solving the time-varying complex QP problems. Besides, the superiorities of complex ZNN models are substantiated, as compared with complex GNN ones.