Zeroing neural networks: A survey (cid:2)

Using neural networks to handle intractability problems and solve complex computation equations is becoming common practices in academia and industry. It has been shown that, although complicated, these problems can be formulated as a set of equations and the key is to ﬁnd the zeros of them. Ze-roing neural networks (ZNN), as a class of neural networks particularly dedicated to ﬁnd zeros of equations


Introduction
Approaches based on neural network for solving various knotty problems have attracted considerable attention in many fields [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . For example, an adaptive fuzzy controller based on neural network is constructed for a class of nonlinear discrete-time systems with discrete-time dead zone in [1] . An adaptive decentralized scheme based on neural network is presented for multipleinput and multiple-output (MIMO) nonlinear systems with the aid of back-stepping techniques in [14] . Such a scheme guarantees the uniform ultimate boundedness of all signals in the closed-loop system with respect to mean square. To overcome the design diffi-culty of nonstrict-feedback structure, Ref. [3] uses variable separation technique to decompose the unknown functions of all state variables into a sum of smooth functions of each error dynamic. With the aid of radial basis function neural networks' universal approximation capability, an adaptive neural control algorithm is proposed in [3] . Authors in [8] propose a neural network model to generate winner-take-all competition, which has an explicit explanation of the competition mechanism. As a branch of artificial intelligence, recurrent neural network (RNN) models have received considerable investigation in many scientific and engineering fields, which is often exploited for computational problems [15][16][17][18][19][20][21][22] and nonlinear optimizations are solved by many methods [23,24] A gradient-based RNN model is presented in [25] for computing the inversion of a matrix online with guaranteed convergence, which can be deemed as a seminal work in this field. A simplified neural network model is presented in [26] to solve a class of linear matrix inequality problems, of which the stability and solvability are analyzed theoretically. In general, recurrent neural networks can be divided into two classes: (1) the continuous-time RNNs and (2) the discrete-time RNNs. By exploiting a numerical differential formula, a continuous-time RNN model can be discretized into a discrete-time one. However, a numerical differentiation rule does not necessarily generate a conver- gent and stable discrete-time RNN model even though the original continuous-time RNN model is convergent. In addition, if the discrete-time RNN model is coded as a serial-processing program and performed on the digital computer, it can be considered as a numerical algorithm [27] . As a novel type of RNN specifically designed for solving time-varying problems, zeroing neural network (ZNN) is able to perfectly track time-varying solution by exploiting the time derivative of time-varying parameters. Then, many researchers make progresses along this direction by proposing various kinds of ZNN models for solving problems with different highlights. A detailed survey and summary are necessary for understanding the development of ZNN models as well as their applications. This paper is organized as follows. In Section 2 , the descriptions and continuous-time ZNN models are presented, which include the evolution of models, activation functions, finite-time convergence, and integration-enhanced ZNN models. In Section 3 , a brief review on the discrete-time ZNN models is presented. In Section 4 , the applications of ZNN techniques are also analyzed. Section 5 concludes this paper with finial remarks.

Design formulas and various continuous-time models
Prior to the proposal of ZNN approach, many gradient-related methods had been reported on the solutions of algebraic equations and optimizations, i.e., zero-finding problems [28][29][30][31] . By constructing a performance index whose minimal point is identical to the solution to the task problem, a typical approach is to design a recurrent neural network evolving along the negative gradient descent to achieve a minimum of the performance index. However, these methods may fail to work well when exploited to the online solution of dynamic problems with time-varying coefficients, which is intrinsically due to the lacking of the compensation to the velocity components of the time-varying coefficients. Therefore, in view of the variability of coefficients, any method designed intrinsically for computing the static problem can no longer guarantee the decrease of the performance index of a time-varying problem, thereby possibly leading to a failure of the task with large residual error. For example, it is observed, investigated and analyzed in [32][33][34] that the residual error of gradient-based neural network (GNN) for solving a time-varying problem can not be eliminated and remains at a relative high level. Refs. [27,35,36] further point out that, when exploited to solve a time-varying problem, any traditional method that does not exploit the time-derivative information of time-varying coefficients can not converge to the theoretic solution with the residual error proportional to the value of the sampling gap.
To solve a time-varying problem in an error-free manner, Zhang et al. present a recurrent neural network for solving the timevarying Sylvester equation, which is depicted in an implicit dynamical system and can be deemed as the seminal work on ZNN [37] . They further generalize and summarize the design procedures of such a methodology, and analyze the convergence and stability of the corresponding ZNN model for time-varying matrix inversion in [38] . Specifically, for solving a time-varying matrix inversion problem depicted in the form of where A (t) ∈ R (n ×n ) is a smooth matrix with its derivative assumed to be known, I ∈ R (n ×n ) is the identity matrix, is the unknown matrix to be obtained.
The core in the design of ZNN model is to construct an error function E(t ) = A (t ) X (t ) − I, which is evidently different from the performance index of gradient-related methods. Then, the ZNN design formula is used to enforce the corresponding E ( t ) to converge to zero: x 4 (t) = a (t) −a (t) [44] Linear system [38] Matrix inversion [45] Matrix square [46] Nonlinear where γ > 0 and ( ·) is a matrix array of activation function φ( ·). Similarly, for the vector-valued time-varying problems [39] , e.g., the system of linear equation Even for the scalar-valued timevarying problems [40][41][42][43] , e.g., the time-varying 4th root finding problem x 4 (t) = a (t ) , with a (t ) ∈ R and x (t) ∈ R , the error function can be designed as e (t) = x 4 (t) − a (t) . Note that the matrixvalued (or vector-valued) error function is a decoupled system and thus its ij th (or i th) subsystem, i.e., the scale-valued dynamical system ˙ e (t) = −γ φ(e (t)) , can be used to analyze the corresponding convergence and stability. By exploiting ZNN design formula to solve different time-varying problems, various ZNN models that exploit the time-derivative information of coefficients can be constructed with their formulations shown in Table 1 . In short, any ZNN model for solving any time-varying problem can be deemed as an equivalently expansion of the ZNN design formula. Since Zhang et al. proposed ZNN models in the 20 0 0s, modified models have been frequently proposed by considering different internal and external factors. Especially, when nonlinear activation functions are incorporated into the network models, stability research has gained significant progress. A brief review on the design of continuous-time ZNN models for various problems solving is presented in [47] . However, with the rapid development of the theory on ZNN, new variations have taken place [48] and in the ensuing part, we will briefly review some basic models of ZNN from different perspective.

Convergence and stability
In the research of neural networks, the key issues are convergence and stability. Broadly speaking, there are three ways for proving the convergence of ZNN models, i.e., proof based on Lyapunov theory, ordinary differential equation (ODE), or Laplace transform.
(1) Proof based on Lyapunov theory [49] . For example, for the time-varying nonlinear minimization problem solving with the task function being f (x (t ) , t ) ∈ R and x (t) ∈ R n in [36] , the error function can be designed as By constructing a Lyapunov function candidate: it can be concluded that V ( t ) is evidently of the positivedefiniteness. Then, computing its time derivative leads to ˙ which is of negative-definiteness and we draw the conclusion that the residual error of the corresponding ZNN model globally converges to zero. It is worth noting that, by replacing the definition of error function e ( t ), the global convergence of all the existing continuous-time ZNN models can be analyzed from the similar way. This way is the dominant approach in the analysis of continuous-time ZNN models and has been widely studied in [36,[50][51][52] .
(2) Proof based on ODE. In addition to the convergent range, ODE-based approach can be used to prove the convergent speed of linear function activated ZNN models. For example, for the same problem shown in [36] , by solving the i th subsystem of design formula, i.e., ˙ where e (0) is the initial value of e ( t ). Then, we have the conclusion that the residual error of the ZNN model globally and exponentially to zero. This way has been widely studied in [36,38,39] .
In view of γ > 0, it can be readily concluded that the final value theorem applies. Based on the final value theorem, we have which completes the proof. This way is a new approach and has been preliminarily studied in [53][54][55][56] .

Nonlinear activation functions
In the design and construction of ZNN models, nonlinear activation functions are used to accelerate the convergent speed. Typically, the following ones are often employed to construct ZNN models [45,57,58] : • the power-sum activation function: • the power-sigmoid activation function: where p is an odd integer and ξ > 0.
• and the hyperbolic sine activation function: where m is an odd integer.
It is worth noting that, to prove the convergence of these nonlinear function activated ZNN models, a common approach is to construct a Lyapunov function candidate V (t) = e T (t ) e (t ) / 2 , and then compute its time derivative ˙ V (t) , which is smaller than that of linear function activated ZNN model. Then, we draw the conclusion that nonlinear activation function can be used to accelerate the convergence speed. Many existing results on the ZNN concern the case that activation functions should be continuous, and strictly monotonically increasing, which is a limitation and should be remedied in the future.

Finite-time convergence
Due to in-depth research on the ZNN model and inspired by the study on finite-time convergence in continuous autonomous system [59][60][61] , Li et al. present a nonlinear function to accelerate the continuous-time ZNN to finite-time convergence for solving timevarying Sylvester equation in [62] and then extend to the online solution of dual neural networks for solving quadratic programming problems in [63] . After that, many different finite-time activation functions have been proposed and employed to various ZNN models, e.g, time-varying matrix pseudoinversion in complex domain [64] , Lyapunov equation [65] , equality-constrained quadratic optimization [66] , linear complex matrix equation [67] and so on [43,[68][69][70] . Note that the residual error of a traditional ZNN model exponentially converges to zero, which means that the smaller the residual error is, the slower the convergent speed is. Therefore, for achieving finite-time convergence, an effective is to amplify the value of ˙ e (t) /e (t) to large enough when e ( t ) approaches to zero. In addition, it is investigated in [71] that various ZNN models can be designed by exploiting different error functions for a time-varying problem solving, which can be accelerated to finite-time convergence.

Complex-valued ZNN models
In the past years, different ZNN models have received considerable studies in many scientific and engineering fields, which successfully tackles the estimation error problem in the real domain. In contrast to these ZNN models defined in the real domain, the research on complex-valued ZNN models shows advantage over conventional real-valued neural networks in complex problems solving. The first complex-valued ZNN model is proposed to solve the time-varying matrix-inversion problems in complex domain in [52] . To guarantee the global convergence of the neural network, only linear activation functions are considered in it. As pointed previously, many nonlinear function can be used to accelerate the convergence speed of ZNN model, which inspires Li et al. to explore nonlinear complex-valued activation functions to accelerate the convergence of ZNN with guaranteed global convergence in [72] . They find two classes of activation functions to achieve the global convergence of the complex-valued ZNN for solving the complex-valued time-varying Sylvester equation and then accelerate it to finite-time convergence. Liao et al. propose five ZNN models by exploiting different techniques to compute the time-varying complex-valued pseudoinversion problem in [73] , which is further generalized to complex-valued matrix inversion in [74] . Qiao et al. present two complex-valued ZNN models for computing the Drazin inverse, which is accelerated to finite-time with proven upper bounds of the convergence time in [69] . In addition, as a comparison, online solution of complex-valued systems of linear equations is investigated in the complex domain via a GNN model in [75] .

Noise-tolerant ZNN models
Many computational models for solving time-varying problems (including the conventional ZNN models) usually assume that the solving task is free of noises or that the noise reduction operation has been completed before the computation [76] . Note that, for the online computation of time-varying problems, time Table 2 Continuous-time noise-tolerant ZNN models constructed for solving timevarying problems.

Dynamic problem
Noise-tolerant ZNN model 4th root finding ˙ Linear system is precious and the preprocessing for denoising may consume extra time, thereby breaking the requirement for online computation. Therefore, an integration-enhanced ZNN design formula is proposed in [55] for time-varying matrix inversion, which leverages the integration-control technique in control theoretical and is able to handle simultaneously the noises. Then, such a noisetolerant ZNN design formula is used to construct continuous-time ZNN model in [56] with nonlinear activation function exploited for accelerating the noise-tolerant model. In [54] , the authors analyze and design different ZNN models via a systematic approach from the control perspective for solving time-varying problems. The essence of the noise-tolerant ZNN models is to leverage the error integration information to eliminate the constant bias errors. A challenging topic existing in this approach is how to determine the suitable activation functions for accelerating its convergence. By exploiting noise-tolerant ZNN design formula to solve different time-varying problems, various noise-tolerant ZNN models that exploit the time-derivative information of coefficients as well as the error integration can be constructed with their formulations shown in Table 2 .

Numerical formulas and various discrete-time models
As reviewed previously, various continuous-time ZNN models have been presented for solving different problems. However, due to the fact that step size in simulating continuous-time systems is variable and that digital computer often requires constant time step, it is different for digital computer to implement continuoustime ZNN models directly [77][78][79][80] . In addition, for continuous-time ZNN models, it is always assumed that neurons communicated and responded instantaneously and without any delay. However, in digital circuits, time delay is unavoidable due to the existing of the sampling gap. Thus, researchers devote their effects to propose and investigate discrete-time ZNN models for online time-varying problems solving. In addition, some challenges of developing the discrete-time ZNN models from the continuous-time ones for the time-varying problems solving are listed as follows.
1. From the perspective of time, any time-varying problem can be deemed as a causal system and thus, the related computation should be conducted based on the existing data, i.e., the present and/or previous data, due to the unavailability of future data but for computing the future solution. For example, for solving the time-varying matrix inversion problem 1 in a discrete manner, at the time instant t k , we can use the known information, e.g., A ( t k ) and ˙ A (t k ) , not the unknown information, e.g., A (t k +1 ) and ˙ A (t k +1 ) for computing the inverse of A (t k +1 ) , i.e., X (t k +1 ) .
Thus, a fundamental requirement for constructing the discretetime ZNN model is that we can not use future data. The first discrete-time ZNN model is proposed in [81] for constant matrix inversion, which bridges the gap between discretetime ZNN model and the traditional Newton iteration. At the early stage, Euler forward difference is used to construct the discretize the continuous-time ZNN model [82,83] . The discrete-time ZNN models generated by Euler forward difference are of the error pattern of O ( τ 2 ), where τ denotes the sampling gap. For example, the Euler-type discrete-time ZNN model for time-varying matrix inversion is directly given as [84] : where k denotes the iteration index and h = τ γ > 0 . In addition, by omitting the term τ X k ˙ A k X k and letting h = 1 , the Euler-type discrete-time ZNN model (4) reduces to which is the Newton iteration. In other words, the traditional Newton iteration can be deemed as a special case of Euler-type discrete-time ZNN model (4) . Meanwhile, the link between Getz-Marsden dynamic system and discrete-time ZNN models is found in [85] . To achieving high accuracy in the discretization of ZNN models, a Taylor-type numerical differentiation formula is proposed in [84] for the first-order derivative approximation, which has a truncation error of O ( τ 2 ) and formulated as A new Taylor-type discrete-time ZNN model can be developed for time-varying matrix inversion as The above Taylor . Note that, the discrete-time ZNN models can be deemed as time delay systems, and thus, a large value of h may lead the model to oscillate. To remedy the instability, a direct way is to lessen the value of h . However, lessening the value of h would significantly slow the convergence of discrete-time ZNN models.

Applications
As a systematic approach for solving time-varying zero-finding problems, ZNN design formula as well as the derived models has been applied to the motion generation and control of redundant manipulators, the tracking control of chaotic systems or even the population control in biosciences. Along the above lines, we will give a detailed review on the application research of ZNN in this section.

Robotic applications
The joint-drift problem existing in the motion generation and control of redundant manipulators is that, after completing a closed path in the workspace, the joint variables do not go back to their initial values [87] . Researchers exploit the ZNN approach to remedy such a weakness. For a redundant robot manipulator, we have [88][89][90][91][92] : where r ∈ R m is the orientation vector to be achieved; f ( θ ) denotes the end-effector position; Differentiating Equation (9) with respect to time t leads to (8) . To eliminate the joint displacement θ (T ) − θ (0) , where θ ( T ) and θ (0) denote the values of joint variables at the final state and initial state, respectively, a direction based on ZNN design formula is to construct an error function (t) = θ (t) − θ (0) and then employ the ZNN design formula to force ( t ) converging to zero. Then, we have ˙ (0)) . For considering the equality and inequality constraints, it is better to minimize the performance index ˙ subject to It is investigated in [93] that the above scheme is reformulated as a quadratic programming problem. Moreover, different RNN models are investigated comparatively for solving such a problem. In general, the redundancy-resolution problem arising in the motion planning and control of redundant manipulators can be solved at the joint-velocity level or at the joint-acceleration level, resulting in the corresponding velocity-level and accelerationlevel redundancy-resolution schemes. Then, to solve the jointangle drift problems in repetitive motion control of redundant robot manipulators at the level of acceleration, a scheme called acceleration-level drift-free (ALDF) scheme subject to a linear equality constraint is presented in [94] with guaranteed effectiveness. Reference [95] point out that there existing equivalent relationship between joint-velocity level and joint-acceleration level via the ZNN-related techniques. Miao et al. present ZNN models for solving time-varying quadratic program problems and applied to robot tracking [96] , where the models are of finite-time convergence. A noise-tolerant ZNN model is designed and presented in [97] for the motion generation of multiple redundant robot manipulators in a distributed manner, which can be deemed as a seminal work on ZNN with application to distributed systems. The Jacobian equality constraints for multiple manipulators in [97] are compacted into an equality based on consensus with limited communications, and the scheme could coordinately control all manipulators involved to complete a given task with the aid of ZNN, as long as the communication topological graph is connected.

Chaos applications
For the tracking-control problems of two chaotic systems, many methods fail to solve it due to the existence of singularities. For demonstration, the Lu system equipped with a single control input u is presented. Let us consider the following chaotic system with input u : where a = 36 , b = 3 and c = 20 ; ϑ = z denotes the output of system (14) . The aim is to construct a controller such that ϑ tracks time-varying desired path ϑ d , with e = ϑ − ϑ d approaching zero.
To adopt ZNN design formula, the first error function is designed as where ϑ d denotes the desired trajectory. In addition, the following ZNN design formula is provided: Expanding the above design formula leads to To get an expression of u in (16) explicitly, the following second error function is constructed: With the aid of (14) , we have where Thus, a controller based on ZNN design formula is obtained: which has a singularity plane x = 0 . To conquer it, the idea based on ZNN-related approach is to eliminate the division operation by transforming the direct control (19) into a minimization problem.
By defining h = x 2 z + (a + b − c − 2 μ) xy − ay 2 − xu + g 2 and based on the gradient method, a tracking controller in the form of ˙ u can be designed as ˙ u = γ xh, (20) which is free of division operation. In view of the fact that such a controller-design method is consisted of ZNN method and GNN method, it is termed ZG neural dynamics in [98] . Such a method is extended to the singularity-conquering of inverted pendulum on a cart system in [99] , the tracking control of chaotic systems with the mixture of additive and multiplicative inputs [100] .

Other applications
In addition to the robotics as well as chaos, ZNN is also applied to the multi-dimensional spectral estimation. For avoiding the computation of direct inverse of covariance matrix, Benchabane et al. use discrete-time ZNN models to compute the inverse of covariance matrix online in [79] . To handle the output tracking control of general-form single-input single-output nonlinear system, controllers based on ZNN are designed in [101] , which can avoid the division-by-zero problem. In [102] , discrete-time ZNN models are used for motion estimation for computing 2D optical flow. In addition, new fractals are yielded by using the discretetime complex-valued ZNN for solving time-varying nonlinear equations in the complex domain in [103][104][105] . The population control of the Lotka-Volterra model in mathematical ecology based on the ZNN-related techniques is investigated in [106] . The presented controller is able to drive the prey population and/or predator population to a desired state.

Conclusion
In summary, the studies for ZNN have achieved a great deal in the last 16 years. However, there are still many new problems to be solved. For future directions of the study on zeroing neural networks as well as their applications, we now provide some prospective suggestions.
1. Continue to apply and find some useful one-step ahead numerical differential formulas to discretize the continuous-time ZNN models, especially to further expand the step-size in the existing discrete ZNN results while keeping a high computational accuracy. This direction is closely related to the development of applied mathematics and computational mathematics. 2. How to construct nonlinear activation functions to accelerate the convergence speed of zeroing neural networks for solving complex-valued problems is still an open problem. For the case of real-valued problems, a great deal of nonlinear activations have been proposed and investigated. Moreover, how to obtain the convergence conditions is also meaningful in the development of zeroing neural networks. 3. In addition to the global stability property, how to construct and analyze the stability criteria for noise-tolerant and nonconvex-allowed zeroing neural networks still needs more effort s. In general, these modified zeroing neural networks are highly related to the control techniques [107,108] .
All these future developments will accompany the development of mathematical theory, especially applied mathematics and computational mathematics. For example, new models should be derived for solving the inequality constraint in time-varying optimization problems. All these future developments will accompany the development of the computational mathematics for constructing and developing neural networks as well as the techniques for various kinds of robot manipulators. Keeping in mind, different kinds of neural networks e.g., discrete-time ZNN models or even the gradient neural network models, have their own feasible ranges, and one cannot expect that only a few existing results on neural networks can tackle all the computational problems.