# Noise-Tolerant Zeroing Neural Dynamics for Solving Hybrid Multilayered Time-Varying Linear Equation System

Authors: Jian Li; Yang Shi; Yange Sun; Jianwei Fan; Mohammed Aquil Mirza
Journal: Security and Communication Networks
 (2022)

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## Abstract

Resource allocation problem in a wireless sensor network can be formulated as time-varying optimization, which can be further converted as time-varying linear equation system (TVLES). Hybrid multilayered time-varying linear equation system (HMTVLES) involving hybrid multilayers and time-variation characteristic is a complicated and challenging problem. Recently it has been solved by zeroing neural dynamics (ZND) method under ideal conditions, i.e., without noises. However, noises are ubiquitous, immanent and unavoidable in real-time systems. In this work, we propose a noise-tolerant zeroing neural dynamics (NTZND) model for solving HMTVLES. It can deal with different kinds of noises such as constant noise, linear-increasing noise, and random noise. Theoretical analyses guarantee the precision of NTZND model in the presence of different kinds of noises. In addition, a general NTZND model is proposed based on general activation function. Besides, classical ZND method and gradient neural dynamics (GND) method are also investigated and compared. Numerical experimental results are presented to verify the theoretical results of proposed models.


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## Body

## 1. Introduction

Noises are ubiquitous, immanent and unavoidable in reality [1–10], such as resource allocation problem in a wireless sensor network [1, 2] and robot control [4–6]. Noises have different forms, such as constant noises, linear noises and random noises. For example, constant noises may exist if the hardware wear results in a fixed deviation. Random noises may exist due to sudden changes in the external environment. Noises existing in linear equation system solving process is necessary to be suppressed as many problems in reality are formulated as linear equation system, in which noises exist [9, 11]. There are many methods to solve classical linear equation system (i.e., time-invariant linear equation system) [12–16], such as decomposition, Gaussian elimination, and Jacobi iteration.The rapid development of industry makes real-time computation increasingly required in industrial engineering. More and more researchers have investigated time-varying systems [17–23], which can better describe real-time feature than time-invariant ones. In [1, 2], resource allocation problem in a wireless sensor network was formulated as time-varying optimization, which can be further converted as time-varying linear equation system (TVLES) [6]. Specifically, the resource allocation problem is formulated as time-varying convex optimization under time-varying linear equality constraints. Then, we use Lagrangian multiplier method, and then the optimization can be converted as TVLES after calculating the partial derivatives. TVLES, which originates from time-invariant linear equation system, has been investigated [24–28]. Zeroing neural dynamics (ZND) method is a good alternative for solving TVLES [11, 21, 27–35]. ZND is also called Zhang neural dynamics and Zhang neural network, which is a special class of recurrent neural network and originates from Hoefeld neural network [21, 27, 28]. In [24], a unified finite-time convergence ZND model was proposed for time-varying linear equations, and the ZND model is applied to robotic applications. In [25], a new ZND model with varying-parameter was proposed to solve time-varying overdetermined system of linear equations. In [27], TVLES with time-varying rank-deficient coefficient was investigated and solved by a new discrete-time ZND model with least-squares solution obtained. In addition, in [11], a noise-suppressing neural model was proposed for solving the time-varying system of linear equations in a control-based approach perspective.Classical time-varying problems are single-layered, which have relatively weak ability to describe practical problems [36–40]. Recently, by introducing multilayers in time-varying problems, multilayered time-varying problems are investigated [41–45]. For example, in [37], multilayered time-varying linear systems including equality and inequality layers were investigated by ZND method. In [39], general discrete-time ZND model with 7-instant was proposed for solving multilayered time-varying system with nonlinear inequality layer and linear equation layer. In [45], multilayered time-varying nonlinear equation system was investigated and solved and the proposed model was further applied to robot manipulator control. In addition, in [36], multilayered time-varying linear equation system (MTVLES) was solved by ZND method.For better describing practical problems, hybrid multilayered time-varying linear equation system (HMTVLES) was investigated in [46], which is generated by adding a hybrid structure into MTVLES. The problem of HMTVLES was investigated and solved by ZND method in [46]. However, most researches about multilayered time-varying problems including HMTVLES do not consider noises and the problem solving is in idea conditions. Inspired by [3–6], in this work, we propose a noise-tolerant zeroing neural dynamics (NTZND) model for solving HMTVLES. It can deal with different kinds of noises such as constant noise, linear-increasing noise and random noise. Theoretical analyses guarantee the precision of NTZND model in the presence of different kinds of noises. In addition, a general NTZND model is proposed based on general activation function. Besides, classical ZND method and gradient neural dynamics (GND) method [47, 48] are also investigated and compared. Numerical experimental results are presented to verify the theoretical results of proposed models. The main contributions of this work are listed as follows:(1)
NTZND model is proposed for solving HMTVLES in the presence of different kinds of noises such as constant noise, linear-increasing noise and random noise.(2)
The effectiveness and precision of NTZND model in the presence of different kinds of noises are analyzed and guaranteed by theoretical analyses.(3)
General NTZND model is proposed based on general activation function. The corresponding theoretical analyses are provided.The remainder of this paper is organized into five sections. Section2 shows preliminaries including problem formulation and classical ZND and GND methods are introduced, and then, NTZND model is proposed to solve the problem of HMTVLES. Section 3 provides some theorems to guarantee the effectiveness and precision of NTZND model in the presence of different kinds of noises. Section 4 proposes a general NTZND model to solve the problem of HMTVLES. Section 5 illustrates numerical experimental results to verify the theoretical results. Section 6 concludes this work with final remarks.


## 2. Preliminaries and Motivations

In this section, preliminaries including problem formulation and classical ZND and GND methods [18, 27] are introduced, and then, NTZND model is proposed to solve the problem of HMTVLES. For the convenience of readers, abbreviations in this paper are summarized in Table 1.Table 1
Abbreviations and their explanations.
AbbreviationExplanationTVLESTime-varying linear equation systemMTVLESMultilayered time-varying linear equation systemHMTVLESHybrid multilayered time-varying linear equation systemZNDZeroing neural dynamicsGNDGradient neural dynamicsNTZNDNoise-tolerant zeroing neural dynamics


### 2.1. Problem Formulation and Definitions

The problem formulation of HMTVLES originates from MTVLES and TVLES. In this subsection, problem formulation and definitions are revisited and shown as follows.Definition 1.
The time-invariant linear equation system [13] is defined as(1)Ax=b,where A∈ℝm×n is a time-invariant full-rank matrix with m≤n; b∈ℝm is a time-invariant vector; and x∈ℝn is unknown and to be solved.Adding time variablet into time-invariant linear equation system (1) leads to TVLES as follows.Definition 2.
The time-varying linear equation system (TVLES) [24] is defined as(2)Atxt=bt,where At∈ℝm×n is a time-varying matrix with m≤n, which is assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved.By introducing two layers on the basis of TVLES (2), in which one layer is with respect toxt and the other is with respect to its time-derivative x.t, the MTVLES is constructed as follows.Definition 3.
The multilayered time-varying linear equation system (MTVLES) [36] is defined as(3)Atxt=bt,Ctx.t=dt,where At∈ℝm1×n and Ct∈ℝm2×n are time-varying matrices with m1+m2≤n, which are assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved, such that two layers of (3) are satisfied.Furthermore, designing a hybrid structure in the second layer of MTVLES (3), we have the HMTVLES as follows.Definition 4.
The hybrid multilayered time-varying linear equation system (HMTVLES) [46] is defined as(4)Atxt=bt,Ctx.t+Etxt=dt,where At∈ℝm1×n, Ct∈ℝm2×n and Et∈ℝm2×n are time-varying matrices with m1+m2≤n, in which the combination of At and Ct is assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved, such that two layers of (4) are satisfied.


### 2.2. ZND and GND Models for HMTVLES

ZND method is an alternative to solve HMTVLES [46]. The constructive process of ZND model is as follows.First, on the basis of the first layer of HMTVLES (4), a vector-form error function is defined as(5)εt≔Atxt−bt.Second, dynamics formula is designed to make error function (5) tend to be zero vector, which means that the designed dynamics formula is applied to every elements of error function. We define any element of error function as εit with i=1,2,…,m1, and dynamics formula is designed as(6)εi.t=−λφεit,where λ is a positive constant, which is utilized to adjust the convergent rate of the proposed models. φ⋅ is the activation function, which must be a monotonically-increasing odd activation function [7]. The following monotonically increasing odd activation functions are frequently used to construct the ZND models.(1)
Linear activation function [7]:(7)φεit=εit.(2)
Power activation function [7]:(8)φεit=εipt,where p≥3 is an odd integer.(3)
Biexponential activation function [7]:(9)φεit=expκεit−exp−κεit,where κ=3.(4)
Power-sigmoid activation function [7]:(10)φεit=εipt,ifεit≥1,1+exp−q1−exp−q1−exp−qεit1+exp−qεit,else,where p≥3 and q>2.Third, substituting error function (5) into dynamics formula6 yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(11)Atx.t=−λφεt−A˙txt+b.t.Finally, combining the equivalency (11) and the second layer of HMTVLES (4) yields ZND model as follows:(12)AtCtx.t=−λφεt9−A˙txt+b.tdt−Etxt.For the convenience of comparison, ZND model with measurement noises is shown as follows:(13)AtCtx.t=−λφεt−A˙txt+b.t+ηtdt−Etxt,where ηt is vector-form measurement noises, such as constant noises, time-varying linear noises, random noises.GND method is an alternative to solve TVLES, while it fails to solve HMTVLES (4) [36, 46]. For the convenience of comparison, the constructive process of GND model to solve HMTVLES (4) is as follows.First, we define an energy function as(14)ϵt=Atxt−bt22.Second, the GND design formula is employed to make the energy function (14) tend to be zero, which is shown as follows:(15)x.t=−λ∂ϵt∂xt,where the parameter λ>0.Third, substituting energy function (14) into GND design (15) yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(16)x.t=−λATtAtxt−bt,where T is the transposition operator.Finally, combining the equivalency (16) and the second layer of HMTVLES (4) yields GND model as follows:(17)ICtx.t=−λATtAtxt−btdt−Etxt.GND model with measurement noises is shown as follows:(18)ICtx.t=−λATtAtxt−bt+ηtdt−Etxt.Note that the matrix on the left side of x.t in GND model (17) belong ℝn+m2×n. GND model (17) could be viewed as overdetermined equation, and thus it may fail to solve HMTVLES (4).


### 2.3. NTZND Model for HMTVLES

Noises are ubiquitous, immanent and unavoidable in real-time systems. Traditional ZND and GND methods can not deal with noises well. In this subsection, we propose NTZND model for solving HMTVLES (4) with different measurement noises. Specifically, on the basis of error function (5), a novel dynamics formula is employed as follows [4, 7]:(19)εi.t=−λεit−γ∫0tεiτdτ.Substituting error function (5) into dynamics (19) yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(20)Atx.t=−λεt−γ∫0tετdτ−A˙txt+b.t.Defining∫0tετdτ as yt, we have(21)I00Aty.tx.t=Atxt−bt−λεt−γyt−A˙txt+b.t,where I∈ℝm1×m1 is identity matrix.Combining (21) and the second layer of HMTVLES (4) yields NTZND model as follows:(22)Ftz.t=gt,where(23)Ft=I00At0Ct,z.t=y.tx.t,and(24)gt=Atxt−bt−λεt−γyt−A˙txt+b.tdt−Etxt.NTZND model (22) with measurement noises is shown as follows:(25)Ftz.t=gt,where(26)gt=Atxt−bt−λεt−γyt−A˙txt+b.t+ηtdt−Etxt.


## 2.1. Problem Formulation and Definitions

The problem formulation of HMTVLES originates from MTVLES and TVLES. In this subsection, problem formulation and definitions are revisited and shown as follows.Definition 1.
The time-invariant linear equation system [13] is defined as(1)Ax=b,where A∈ℝm×n is a time-invariant full-rank matrix with m≤n; b∈ℝm is a time-invariant vector; and x∈ℝn is unknown and to be solved.Adding time variablet into time-invariant linear equation system (1) leads to TVLES as follows.Definition 2.
The time-varying linear equation system (TVLES) [24] is defined as(2)Atxt=bt,where At∈ℝm×n is a time-varying matrix with m≤n, which is assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved.By introducing two layers on the basis of TVLES (2), in which one layer is with respect toxt and the other is with respect to its time-derivative x.t, the MTVLES is constructed as follows.Definition 3.
The multilayered time-varying linear equation system (MTVLES) [36] is defined as(3)Atxt=bt,Ctx.t=dt,where At∈ℝm1×n and Ct∈ℝm2×n are time-varying matrices with m1+m2≤n, which are assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved, such that two layers of (3) are satisfied.Furthermore, designing a hybrid structure in the second layer of MTVLES (3), we have the HMTVLES as follows.Definition 4.
The hybrid multilayered time-varying linear equation system (HMTVLES) [46] is defined as(4)Atxt=bt,Ctx.t+Etxt=dt,where At∈ℝm1×n, Ct∈ℝm2×n and Et∈ℝm2×n are time-varying matrices with m1+m2≤n, in which the combination of At and Ct is assumed to be always full-rank over time; bt∈ℝm is a time-varying vector; and xt∈ℝn is unknown and to be solved, such that two layers of (4) are satisfied.


## 2.2. ZND and GND Models for HMTVLES

ZND method is an alternative to solve HMTVLES [46]. The constructive process of ZND model is as follows.First, on the basis of the first layer of HMTVLES (4), a vector-form error function is defined as(5)εt≔Atxt−bt.Second, dynamics formula is designed to make error function (5) tend to be zero vector, which means that the designed dynamics formula is applied to every elements of error function. We define any element of error function as εit with i=1,2,…,m1, and dynamics formula is designed as(6)εi.t=−λφεit,where λ is a positive constant, which is utilized to adjust the convergent rate of the proposed models. φ⋅ is the activation function, which must be a monotonically-increasing odd activation function [7]. The following monotonically increasing odd activation functions are frequently used to construct the ZND models.(1)
Linear activation function [7]:(7)φεit=εit.(2)
Power activation function [7]:(8)φεit=εipt,where p≥3 is an odd integer.(3)
Biexponential activation function [7]:(9)φεit=expκεit−exp−κεit,where κ=3.(4)
Power-sigmoid activation function [7]:(10)φεit=εipt,ifεit≥1,1+exp−q1−exp−q1−exp−qεit1+exp−qεit,else,where p≥3 and q>2.Third, substituting error function (5) into dynamics formula6 yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(11)Atx.t=−λφεt−A˙txt+b.t.Finally, combining the equivalency (11) and the second layer of HMTVLES (4) yields ZND model as follows:(12)AtCtx.t=−λφεt9−A˙txt+b.tdt−Etxt.For the convenience of comparison, ZND model with measurement noises is shown as follows:(13)AtCtx.t=−λφεt−A˙txt+b.t+ηtdt−Etxt,where ηt is vector-form measurement noises, such as constant noises, time-varying linear noises, random noises.GND method is an alternative to solve TVLES, while it fails to solve HMTVLES (4) [36, 46]. For the convenience of comparison, the constructive process of GND model to solve HMTVLES (4) is as follows.First, we define an energy function as(14)ϵt=Atxt−bt22.Second, the GND design formula is employed to make the energy function (14) tend to be zero, which is shown as follows:(15)x.t=−λ∂ϵt∂xt,where the parameter λ>0.Third, substituting energy function (14) into GND design (15) yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(16)x.t=−λATtAtxt−bt,where T is the transposition operator.Finally, combining the equivalency (16) and the second layer of HMTVLES (4) yields GND model as follows:(17)ICtx.t=−λATtAtxt−btdt−Etxt.GND model with measurement noises is shown as follows:(18)ICtx.t=−λATtAtxt−bt+ηtdt−Etxt.Note that the matrix on the left side of x.t in GND model (17) belong ℝn+m2×n. GND model (17) could be viewed as overdetermined equation, and thus it may fail to solve HMTVLES (4).


## 2.3. NTZND Model for HMTVLES

Noises are ubiquitous, immanent and unavoidable in real-time systems. Traditional ZND and GND methods can not deal with noises well. In this subsection, we propose NTZND model for solving HMTVLES (4) with different measurement noises. Specifically, on the basis of error function (5), a novel dynamics formula is employed as follows [4, 7]:(19)εi.t=−λεit−γ∫0tεiτdτ.Substituting error function (5) into dynamics (19) yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(20)Atx.t=−λεt−γ∫0tετdτ−A˙txt+b.t.Defining∫0tετdτ as yt, we have(21)I00Aty.tx.t=Atxt−bt−λεt−γyt−A˙txt+b.t,where I∈ℝm1×m1 is identity matrix.Combining (21) and the second layer of HMTVLES (4) yields NTZND model as follows:(22)Ftz.t=gt,where(23)Ft=I00At0Ct,z.t=y.tx.t,and(24)gt=Atxt−bt−λεt−γyt−A˙txt+b.tdt−Etxt.NTZND model (22) with measurement noises is shown as follows:(25)Ftz.t=gt,where(26)gt=Atxt−bt−λεt−γyt−A˙txt+b.t+ηtdt−Etxt.


## 3. Theoretical Results and Analyses

Theorem 1.
Assume thatAt∈ℝm1×n, Ct∈ℝm2×n and Et∈ℝm2×n are time-varying matrices with m1+m2≤n, in which the combination of At and Ct is assumed to be always full-rank over time, when the solution of NTZND model (22) starts from any initial state x0, residual error defined as Atxt−bt2+Ctx.t+Etxt−dt2 globally converges to zero as t⟶+∞, where ⋅2 denotes the 2-norm of vectors.Proof 1.
See Appendix A for details. □Theorem 2.
Residual error of NTZND model (22) for solving HMTVLES (4) exponentially converges to zero ast⟶+∞.Proof 1.
See Appendix B for details. □Theorem 3.
Residual error of NTZND model (22) for solving HMTVLES (4) in the presence of unknown vector-form constant noiseηt=η∈ℝm1, i.e., model (23), converges to zero as t⟶+∞.Proof 3.
See Appendix C for details. □Theorem 4.
Residual error of NTZND model (22) for solving HMTVLES (4) in the presence of unknown vector-form linear-increasing noiseηt=ηt∈ℝm1, i.e., model (23), converges to η/γ as t⟶+∞.Proof 4.
See Appendix D for details. □Theorem 5.
Residual error of NTZND model (22) for solving HMTVLES (4) in the presence of unknown vector-form random noiseηt∈ℝm1, i.e., model (23), is approximately in inverse proportion to λ as t⟶+∞.Proof 5.
See Appendix E for details. □


## 4. General NTZND Model

To develop NTZND model, the following dynamics formula is employed:(27)εi.t=−λεit−γ∫0tεiτdτ,in which we simply use linear activation function. More other activation functions can be employed, such as power activation function, biexponential activation function, power-sigmoid activation function. Any monotonically increasing odd activation functions can be used. Thus, we define general monotonically increasing odd activation function as φ⋅ [7]. The general dynamics formula is(28)εi.t=−λφεit−γ∫0tεiτdτ,Substituting error function (5) into general dynamics formula 24 yields the equivalency equation of the first-layer of HMTVLES (4) as follows:(29)Atx.t=−λφεt−γ∫0tετdτ−A˙txt+b.t.Finally, general NTZND model is obtained as follows:(30)Ftz.t=gt,where(31)Ft=I00At0Ct,z.t=y.tx.t,and(32)gt=Atxt−bt−λφεt−γyt−A˙txt+b.tdt−Etxt.The convergence of general NTZND model (26) is proved in the following theorem.Theorem 6.
Assume thatAt∈ℝm1×n, Ct∈ℝm2×n and Et∈ℝm2×n are time-varying matrices with m1+m2≤n, in which the combination of At and Ct is assumed to be always full-rank over time and that φ⋅ is monotonically-increasing odd activation function, when the solution of general NTZND model (26) starts from any initial state xt0, residual error globally converges to zero as t⟶+∞.Proof 6.
See Appendix F for details. □


## 5. Numerical Results and Verifications

In this section, numerous numerical results are presented to verify the theoretical results shown as aforementioned. Codes can be found here athttps://github.com/lijcit/1511895678. Specifically, we take HMTVLES (4) with At∈ℝ4×9, bt∈ℝ4, xt∈ℝ9, Ct∈ℝ4×9, dt∈ℝ4 as an example. By defining each elements of matrices and vectors as Ai,jt, bi,jt, Cit, Ei,jt and dit (i=1,2,3,4 and j=1,…,9), their expressions are as follows:(33)Ai,jt=sin0.1i−jtj−i,wheni>j,sin0.1it+3,wheni=j,cos0.1j−itj−i,wheni<j,bi0t=cost,wheniis odd,sin0.5t,wheniis even,Ci,jt=−sin0.1i+4−jti−j+4,wheni+4>j,sin0.6t+3,wheni+4=j,cos0.1j−i−4tj−i−4,wheni+4<j,Ei,jt=sin0.1i−jtj−i,wheni>j,sin0.1it+3,wheni=j,cos0.1j−itj−i,wheni<j,(34)dit=cos0.5t,wheniis odd,sint,wheniis even.This HMTVLES satisfies the assumptions thatm1+m2≤n with m1=m2=4,n=9 and that combination of At and Ct is always full-rank over time.First, to illustrate the effectiveness of NTZND model (22) without noise, a group of numerical experiments about NTZND model (22) with different values of parametersλ and γ are conducted. The corresponding numerical results are shown in Figure 1. It is observed that, by randomly setting initial state x0, norm-form residual errors of NTZND model (22) with with different values of parameters all exponentially converge to zero. The results verify the theoretical results of Theorems 1 and 2. Besides, solution trajectories generated by NTZND model (22) are also presented.Figure 1
Solution trajectories and residual errors defined asAtxt−bt2+Ctx.t+Etxt−dt2 when using NTZND model (22) to solve HMTVLES (4) with different values of λ and γ. (a) Solution trajectories. (b) Residual error with different values of λ. (c) Residual error with different values of γ.
(a)(b)(c)Second, to illustrate the performances and superiority of NTZND model (22) in the presence of constant noise, i.e., model (23), different constant noises are added in to the solution process. For comparison, ZND model (13) and GND model (18) are also employed. Numerical results are shown in Figure2. It is observed that, when different values of constant noises are added, NTZND model (23) performs well and converges to zero. In contrast, ZND model (13) and GND model (18) fail to deal with constant noises and even diverge with large constant noise. The numerical results coincide with the results in Theorem 3.Figure 2
Residual errors when using NTZND model (23), ZND model (13) and GND model (18) to solve HMTVLES (4) in the presence of different constant noises. (a) With noiseη=1. (b) With noise η=10. (c) With noise η=100.
(a)(b)(c)Third, to illustrate the performances and superiority of NTZND model (22) in the presence of linear-increasing noise, i.e., model (23), linear-increasing noises are added into the solution process. Numerical results are shown in Figure3. It is observed that, NTZND model (23) performs well while ZND model (13) and GND model (18) fail to deal with linear-increasing noises. In addition, to illustrate the relationship of solution precision and parameter γ, different values of γ are considered. From the numerical result, the solution precision is approximately inversely proportional to parameter γ, which coincides with the result in Theorem 4.Figure 3
Residual errors when using NTZND model (23), ZND model (13) and GND model (18) to solve HMTVLES (4) in the presence of linear-increasing noise with different values ofγ. (a) With γ=10. (b) With γ=100. (c) With γ=1000.
(a)(b)(c)Fourth, to illustrate NTZND model (22) in the presence of random noise, i.e., model (23), random noises are added in to the solution process. Numerical results are shown in Figure4. NTZND model (23) still performs well while GND model (18) fails to deal with linear-increasing noises. ZND model (13) can deal with random noise although the precision is lower than model (23). In addition, to illustrate the relationship of solution precision and parameter λ, different values of λ are considered. From the numerical result, the solution precision is approximately inversely proportional to parameter λ, which coincides with the result in Theorem 5.Figure 4
Residual errors when using NTZND model (23), ZND model (13) and GND model (18) to solve HMTVLES (4) in the presence of random noise with different values ofλ. (a) With λ=10. (b) With λ=100. (c) With λ=1000.
(a)(b)(c)Finally, to illustrate general NTZND model (26) in the presence of different kinds of noises, more numerical experiments about general NTZND model (26) with different activation functions are conducted, and numerical results are shown in Figure5. It is observed that a group of models generated by general NTZND model (26) with different activation functions exponentially converge in the presence of different kinds of noises, which coincides with the result in Theorem 6.Figure 5
Residual errors when using general NTZND model (26) with different activation functions to solve HMTVLES (4) in the presence of different kinds of noises. (a) Constant noise. (b) Linear-increasing noise. (c) Random noise.
(a)(b)(c)


## 6. Conclusions

In this work, NTZND model (22) has been proposed to solve HMTVLES (4) in the presence of different kinds of noises (i.e., constant noises, linear-increasing noises and random noises). Classical ZND model (13) and GND model in the presence of different kinds of noises (18) have been introduced and investigated to substantiate the superiority of proposed NTZND model (22). Theoretical analyses have been provided to guarantee the effectiveness and precision of proposed model. Furthermore, general NTZND model (26) has been proposed based on general activation function.



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