Research article

Advancements in enhancing cyber-physical system security: Practical deep learning solutions for network traffic classification and integration with security technologies

  • Traditional network analysis frequently relied on manual examination or predefined patterns for the detection of system intrusions. As soon as there was increase in the evolution of the internet and the sophistication of cyber threats, the ability for the identification of attacks promptly became more challenging. Network traffic classification is a multi-faceted process that involves preparation of datasets by handling missing and redundant values. Machine learning (ML) models have been employed to classify network traffic effectively. In this article, we introduce a hybrid Deep learning (DL) model which is designed for enhancing the accuracy of network traffic classification (NTC) within the domain of cyber-physical systems (CPS). Our novel model capitalizes on the synergies among CPS, network traffic classification (NTC), and DL techniques. The model is implemented and evaluated in Python, focusing on its performance in CPS-driven network security. We assessed the model's effectiveness using key metrics such as accuracy, precision, recall, and F1-score, highlighting its robustness in CPS-driven security. By integrating sophisticated hybrid DL algorithms, this research contributes to the resilience of network traffic classification in the dynamic CPS environment.

    Citation: Shivani Gaba, Ishan Budhiraja, Vimal Kumar, Aaisha Makkar. Advancements in enhancing cyber-physical system security: Practical deep learning solutions for network traffic classification and integration with security technologies[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 1527-1553. doi: 10.3934/mbe.2024066

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  • Traditional network analysis frequently relied on manual examination or predefined patterns for the detection of system intrusions. As soon as there was increase in the evolution of the internet and the sophistication of cyber threats, the ability for the identification of attacks promptly became more challenging. Network traffic classification is a multi-faceted process that involves preparation of datasets by handling missing and redundant values. Machine learning (ML) models have been employed to classify network traffic effectively. In this article, we introduce a hybrid Deep learning (DL) model which is designed for enhancing the accuracy of network traffic classification (NTC) within the domain of cyber-physical systems (CPS). Our novel model capitalizes on the synergies among CPS, network traffic classification (NTC), and DL techniques. The model is implemented and evaluated in Python, focusing on its performance in CPS-driven network security. We assessed the model's effectiveness using key metrics such as accuracy, precision, recall, and F1-score, highlighting its robustness in CPS-driven security. By integrating sophisticated hybrid DL algorithms, this research contributes to the resilience of network traffic classification in the dynamic CPS environment.



    It is well known that the general approaches in fuzzy differential systems are established using the concept of fuzzy sets, as introduced by Zadeh [1] in 1965. The theory of fuzzy sets and some of their applications have been developed in a number of books and papers. We will refer the readers to [2,3,4] and the references cited there.

    On the other hand, the notions of an HHukuhara derivative and H differentiability were introduced in 1983 for fuzzy mappings [5], and the notion of integrals in [6]. Since then, the investigations of fuzzy differential equations have undergone rapid development. See, for example, [7,8,9,10,11,12,13,14] and the references therein. The theory of fuzzy differential equations is still a hot topic for research [15,16,17,18,19] including numerous applications considering fuzzy neural networks and fuzzy controllers [20,21,22,23,24,25].

    It is also well known that the functionality of many complex engineering systems, as well as, the long life of their practical operation are provided under the conditions of uncertainties [26]. In fact, due to inaccuracy in the measurements of the model parameters, data input and different types of unpredictability, uncertain parameters occur in a real system [27]. It is, therefore, clear that the study of the effects of uncertain values of the parameters on the fundamental and qualitative behavior of a system is of significant interest for theory and applications [28,29,30,31,32,33,34], including fuzzy modeling [35]. Considering the high importance of considering uncertain parameters, the theory of uncertain fuzzy differential equations needs future development and this is the basic aim and contribution of our research.

    This paper deals with systems of fuzzy differential equations that simulate the perturbed motion of a system with uncertain values of parameters that belong to a certain domain. A regularization procedure is proposed for the family of differential equations under consideration with respect to the uncertain parameter. An analysis of some fundamental properties of the solutions is performed for both, the original fuzzy system of differential equations and the intermediate families of differential equations. The introduced regularization scheme expands the horizon for the extension of the fundamental and qualitative theory of fuzzy differential equations to the equations involving uncertain parameters. The authors expect that the proposed results will be of particular interest to researchers in the study of the qualitative properties of such systems, including equations with delays and some modifications of the Lyapunov theory. In addition, the engineering applications of such results to the design of efficient fuzzy controllers are numerous.

    The innovation and practical significance of our research are as follows:

    (ⅰ) we introduce a new regularization scheme for uncertain fuzzy differential equations;

    (ⅱ) new existence criteria of the solutions of the regularized fuzzy differential equations are proposed;

    (ⅲ) the distance between two solutions is estimated;

    (ⅳ) the offered regularization scheme reduces a family of fuzzy differential equations to a simple form that allows analysis of the properties of solutions of both the original fuzzy system of differential equations, as well as the intermediate families of differential equations.

    The rest of the paper is organized according to the following scenario. In Section 2 we provide the necessary preliminary notes from the theory of fuzzy sets. Some main properties of fuzzy functions and HHukuhara derivatives are also given. In Section 3 we introduce a regularization procedure for uncertain fuzzy systems of differential equations, and the basic problem of analyzing such systems based on the developed procedure is presented. In Section 4 conditions for existence of the solutions of the regularized fuzzy differential equations are established. In Section 5 we offer an estimate of the distance between solutions of the regularized equations. The closing Section 6 provides some comments and future directions for research.

    This section is mainly based on the results in [2,4,14].

    The use of fuzzy sets introduced in [1] has facilitated the mathematical modeling and analysis of real processes that involve uncertain parameters. Recently, the fuzzy set approach achieved significant development. In this section, we will present some elements from the fuzzy set theory and fuzzy functions that are necessary for the analysis of uncertain systems.

    Consider a basic set X with elements of an arbitrary nature. To each element xX a value of a membership function ξ(x) is assigned, and the function ξ(x) takes its values from the closed interval [0,1].

    Following [1], for a function ξ:X[0,1], we consider a fuzzy subset of the set X as a nonempty subset with elements {(x,ξ(x)):xX} from X×[0,1].

    For a fuzzy set with a membership function ξ on X its ϰ-level sets [ξ]ϰ are defined by

    [ξ]ϰ={xX:  ξ(x)ϰ}for anyϰ(0,1].

    The closure of the union of all ϰ-level sets for a fuzzy set with a membership function ξ in the general topological space X is called its support, and it is denoted by [ξ]0, i.e.,

    [ξ]0=¯ϰ(0,1][ξ]ϰ.

    Note that most often, the space X is the N-dimensional Euclidean space RN equipped with a norm .

    Next, the Hausdorff distance between two nonempty subsets U and V of RN is defined as

    ΔH(U,V)=min{r0:U{VVr(0)},V{UVr(0)}},

    where Vr(0)={xRN:x<r}, r0.

    The above distance is symmetric with respect to both subsets U and V. For more properties of ΔH(U,V) we refer the reader to [2,4,14].

    We will next need the space EN of functions ξ:RN[0,1], which satisfy the following conditions (cf. [2]):

    1) ξ is upper semicontinuous;

    2) there exists an x0RN such that ξ(x0)=1;

    3) ξ is fuzzy convex, i. e.,

    ξ(νx+(1ν)y)min[ξ(x),ξ(y)]

    for any values of ν[0,1];

    4) the closure of the set {xRN:ξ(x)>0} is a compact subset of RN.

    It is well known that [7,8,9,10,11,12,13,14] and [15,16,17,18,19], if a fuzzy set with a membership function ξ is a fuzzy convex set, then [ξ]ϰ is convex in RN for any ϰ[0,1].

    Since EN is a space of functions ξ:RN[0,1], then a metric in it can be determined as

    Δ(ξ,η)=sup{|ξ(x)η(x)|: xRN}.

    The least upper bound of ΔH on EN is defined by

    d[ξ,η]=sup{ΔH([ξ]ϰ,[η]ϰ): ϰ[0,1]}

    for ξ,ηEN. The above defined d[ξ,η] satisfies all requirements to be a metric in EN [2].

    Consider a compact T=[α,β], β>α>0.

    The mapping F:TEN is strictly measurable, if for any ϰ[0,1] the multivalued mapping Fϰ:TPk(RN), defined as Fϰ(τ)=[F(τ)]ϰ, is measurable in the sense of Lebesgue under the condition that Pk(RN) is equipped with a topology generated by the Hausdorff metric, where Pk(RN) denotes the family of all nonempty compact convex subsets of RN [11,36].

    The mapping F:TEN is integrally bounded if there exists an integrable function ω(τ) such that xω(τ) for xF0(τ).

    We will denote by βαF(τ)dτ the integral of F on the interval T defined as

    TF(τ)dτ={Tˉf(τ)dtˉf: IRN is a measurable selection forFϰ}

    for any 0<ϰ1.

    The strictly measurable and integrally bounded mapping F:TRN is integrable on I, if TF(τ)dτEN.

    Numerous important properties of fuzzy functions are given in [2,4,11].

    Let ξ,ηEN. If there exists ζEN such that ξ=η+ζ, ζ is determined as the Hukuhara-type difference of the subsets ξ and η and is denoted by ξη.

    If both limits in the metric space (EN,Δ)

    lim{[F(τ0+h)F(τ0)]h1:h0+} and lim{[F(τ0)F(τ0h)]h1:h0+}

    exist and are equal to L, then the mapping F:TEN is differentiable at the point τ0T, and L=F(τ0)EN.

    The family {DHFϰ(τ):ϰ[0,1]} determines an element F(τ)EN. If Fϰ is differentiable, then the multivalued mapping Fϰ is differentiable in the sense of Hukuhara for all ϰ[0,1] and

    DHFϰ(τ)=[F(τ)]ϰ,

    where DHFϰ is the Hukuhara-type derivative of Fϰ.

    Several basic properties of differentiable mappings F:IEN in the Hukuhara sense are given below:

    1) F is continuous on T;

    2) For τ1,τ2T and τ1τ2 there exists an ιEN such that F(τ2)=F(τ1)+ι;

    3) If the derivative F is integrable on T, then

    F(σ)=F(α)+σαF(τ)dτ;

    4) If θ0EN and θ0(x)={1,forx=0,0,xRN{0}, then

    Δ(F(β),F(α))(βα)supτTΔ(F(τ),θ0). (2.1)

    In this section, we will introduce a regularization scheme for a system of fuzzy differential equations with respect to an uncertain parameter.

    Consider the following fuzzy system with an uncertain parameter

    dξdτ=f(τ,ξ,μ),ξ(τ0)=ξ0, (3.1)

    where τ0R+, ξEN, fC(R+×EN×S,EN), μS is an uncertain parameter, S is a compact set in Rl.

    The parameter vector μ that represents the uncertainty in system (3.1) may vary in nature and can represent different characteristics. More precisely, the uncertainty parameter μ:

    (a) may represent an uncertain value of a certain physical parameter;

    (b) may describe an estimate of an external disturbance;

    (c) may represent an inaccurate measured value of the input effect of one of the subsystems on the other one;

    (d) may represent some nonlinear elements of the considered mechanical system that are too complicated to be measured accurately;

    (e) may be an indicator of the existence of some inaccuracies in the system (3.1);

    (f) may be a union of the characteristics (a)–(e).

    Denote

    fm(τ,ξ)=¯coμSf(τ,ξ,μ),SRl, (3.2)
    fM(τ,ξ)=¯coμSf(τ,ξ,μ),SRl (3.3)

    and suppose that fm(τ,ξ), fM(τ,ξ)C(R+×EN,EN). It is clear that

    fm(τ,ξ)f(τ,ξ,μ)fM(τ,ξ) (3.4)

    for (τ,ξ,μ)R+×EN×S.

    For (τ,η)R+×EN and 0ϰ1, we introduce a family of mappings fϰ(τ,η) by

    fϰ(τ,η)=fM(τ,η)ϰ+(1ϰ)fm(τ,η) (3.5)

    and, we will introduce a system of fuzzy differential equations corresponding to the system (3.1) as

    dηdτ=fϰ(τ,η),η(τ0)=η0, (3.6)

    where fϰC(T×EN,EN), T=[τ0,τ0+a], τ00, a>0, ϰ[0,1].

    We will say that the mapping η:TEN is a solution of (3.6), if it is weakly continuous and satisfies the integral equation

    η(τ)=η0+ττ0fϰ(σ,η(σ))dσ (3.7)

    for all τT and any value of ϰ[0,1].

    The introduced fuzzy system of differential equations (3.6) is considered as a regularized system of the system (3.1) with respect to the uncertain parameter.

    It is clear that for all τT we have diam[ξ(τ)]ϰdiam[ξ0]ϰ, for any value of ϰ[0,1], where diam denotes the set diameter of any level [2,4,11].

    The main goal and contribution of the present paper are to investigate some fundamental properties of the regularized system (3.6) on T and [τ0,).

    In this section we will state criteria for the existence and uniqueness of the solutions of the introduced regularized problem (3.6).

    Theorem 4.1. If the family fϰ(τ,η)C(T×EN,EN) for any ϰ[0,1] and there exists a positive constant Lϰ such that

    d[fϰ(τ,η),fϰ(τ,ˉη)]Lϰd[η,ˉη], τT,η,ˉηEN,

    then for the problem (3.6) there exists a unique solution defined on T for any ϰ[0,1].

    Proof. We define a metric in C(T,EN) as:

    H[η,ˉη]=supTd[η(τ),ˉη(τ)]eλτ

    for all η,ˉηEN, where λ=2maxLϰ, ϰ[0,1]. The completeness of (EN,d)implies the completeness of the space (C(T,EN),H).

    Let ξϰC(T,EN) for any ϰ[0,1] and the mapping Tξϰ is defined as

    Tξϰ(τ)=ξ0+ττ0fϰ(σ,ξϰ(σ))dσ,ϰ[0,1].

    From the above definition we have that TξϰC(T,EN) and

    d[Tη(τ),Tˉη(τ)]=d[η0+ττ0fϰ(σ,η(σ))dσ,η0+ττ0fϰ(σ,ˉη(σ))dσ]
    =d[ττ0fϰ(σ,η(σ))dσ,ττ0fϰ(σ,ˉη(σ))dσ]
    ττ0d[fϰ(σ,η(σ)),fϰ(σ,ˉη(σ))]dσ<maxϰLϰττ0d[η(σ),ˉη(σ)]dσ,τT, ϰ[0,1].

    Since d[η,ˉη]=H[η,ˉη]eλτ, we have

    eλτd[Tη(τ),Tˉη(τ)]<maxϰLϰeλτH[η,ˉη]ττ0eλσdσmaxϰLϰλH[η,ˉη]. (4.1)

    Given the choice of λ, from (4.1) we obtain

    H[Tη,Tˉη]<12H[η,ˉη].

    The last inequality implies the existence of a unique fixed point ξϰ(τ) for the operator Tξϰ which is the solution of the initial value problem (3.6) for ϰ[0,1].

    Now, let us define a second metric in C(T,EN) as follows:

    H[η,ˉη]=supTd[η(τ),ˉη(τ)],

    where η,ˉηEN, and consider a family of continuous functions that have equal variation over a given neighborhood. Such a family of functions is know to be equi-continuous [11].

    Theorem 4.2. If the family fϰ(τ,η)C(T×EN,EN) for any ϰ[0,1] and there exists a positive constant Ωϰ>0 such that

    d[fϰ(τ,η),θ0]Ωϰ, τT, η,θ0EN,

    then for the problem (3.6) there exists a unique solution defined on T for any ϰ[0,1].

    Proof. Let the set B, BC(T,EN) be bounded. Then, according to the definition of the mapping T, the set TB={Tξϰ:ξϰB,ϰ[0,1]} is bounded, if it is equi-continuous, and for any τT the set [TB](τ)={[Tξϰ](τ):τT,ϰ[0,1]} is a bounded subset of EN.

    For τ1<τ2T and ηB, we have from (2.1) that

    d[Tη(τ1),Tη(τ2)]|τ2τ1|maxTd[f(τ,η(τ)),θ0]|τ2τ1|Ωϰ<|τ2τ1|¯Ω, (4.2)

    where ¯Ω=maxϰΩϰ. This implies the equi-continuity of the set TB.

    Also, for any fixed τT we have that

    d[Tη(τ),Tη(τ1)]<|ττ1|¯Ω (4.3)

    for τ1T, ηB.

    From the above inequality, we conclude that the set {[Tξϰ](τ):ξϰB,ϰ[0,1]} is bounded in the space EN, which, according to the Arzela–Ascoli theorem, implies that the set TB is a relatively compact subset of C(T,EN).

    Next, for ϰ[0,1] and M>0 we define B={ξϰC(T,EN):H[ξϰ,θ0]<M¯Ω, B(C(T,EN),H).

    Obviously, TBB, since ξϰC(T,EN), ϰ[0,1] and d[Tξϰ(τ),Tξϰ(τ0)]=d[Tξϰ(τ),θ0]|ττ0|Ωϰ<M¯Ω, ϰ[0,1]. Let θ0(τ)=θ0 for τT, where θ0(τ):TEN. Then

    H[Tξϰ,Tθ0]=supTd[(Tξϰ)(τ),(Tθ0)(τ)]|ττ0|Ωϰ<M¯Ω.

    Hence, T is compact. Therefore, it has a fixed point ξϰ(t), and according to the definition of T, ξϰ(t) is the solution of the initial value problem (3.6) for ϰ[0,1]. This completes the proof.

    Remark 4.3. Theorems 4.1 and 4.2 offered new existence criteria for the regularized system (3.6). The proposed criteria show that the idea to use a family of mappings and regularize the fuzzy system (3.1) with respect to uncertain parameters greatly benefits its analysis. Note that due to some limitations and difficulties in the study of fuzzy differential systems with uncertain parameters, the published results in this direction are very few [11,18]. Hence, the proposed regularization procedure complements such published accomplishments and, due to the offered advantages is more appropriate for applications.

    Remark 4.4. The proposed existence results also extend and generalize some recently published existence results for differential systems with initial and nonlocal boundary conditions [37], where the fixed-point argument plays a crucial role in manipulating the integral equation, to the fuzzy case.

    The validity of Theorem 4.1 is demonstrated by the next example.

    Example 4.5. Let fϰ(τ,η)=Aϰη+Bϰ for any value of ϰ[0,1] and Aϰ,BϰE1. Consider the initial value problem

    dηdτ=Aϰη+Bϰ, (4.4)
    η(τ0)=η0D0E1. (4.5)

    It is easy to show that

    AϰηAϰˉη∣≤Lϰd[η,ˉη],

    where Lϰ=maxAϰ for ϰ[0,1].

    Therefore, all conditions of Theorem 4.1 are satisfied, and hence there exists a unique solution of the initial value problem (4.4)-(4.5). The unique solution is of the type

    η(τ,τ0,η0)=ϰ[0,1][ηϰ(τ,τ0,η0),η0D0], (4.6)

    where

    ηϰ(τ,τ0,η0)=η0expAϰ(ττ0)+(BϰAϰ)[expAϰ(ττ0)1]

    for any value of ϰ[0,1]. A solution in the form of (4.6) is compact for all tT and the union contains all upper and lower solutions of the problem (4.4)-(4.5). For the rationale for this approach, see [11], pp. 150–155).

    Remark 4.6. The conditions of Theorem 4.1 imply the following estimate

    d[fϰ(τ,η),fϰ(τ,ˉη)]d[fϰ(τ,η),θ0]+[θ0,fϰ(τ,ˉη]2Ωϰ

    for all tT, η,ˉηE1 and any value of ϰ[0,1]. Hence, for Ωϰ=12Lϰd[η,ˉη] all conditions of Theorem 4.1. are also satisfied. Therefore, the conditions of Theorem 4.2 are some modifications of these of Theorem 4.1.

    Given that the inequalities

    fm(τ,η)fϰ(τ,η)fM(τ,η),ϰ[0,1]

    are satisfied for all (τ,η)T×EN, it is important and interesting to evaluate the distance between two solutions η(τ), ˉη(τ) of the regularized system (3.6) depending on the initial data. This is the aim of the present section.

    Theorem 5.1. Assume the following:

    1) The family fϰC(T×EN,EN) for any ϰ[0,1].

    2) There exists a continuous function g(τ,ζ), g: T×R+R, which is nondecreasing with respect to its second variable ζ for any τT, and such that for (τ,η),(τ,ˉη)T×EN and ϰ[0,1],

    d[fϰ(τ,η),fϰ(τ,ˉη)]g(τ,d[η,ˉη]).

    3) The maximal solution u(τ,τ0,y0) of the scalar problem

    dy/dτ=g(τ,y),y(τ0)=y00

    exists on T.

    4) The functions η(τ) and ˉη(τ) are any two solutions of the problem (3.6) defined on T, corresponding to initial data (η0,ˉη0) such that d[η0,ˉη0]y0.

    Then,

    d[η(τ),ˉη(τ)]u(τ,τ0,y0), τT. (5.1)

    Proof. Set d[η(τ),ˉη(τ)]=ρ(τ). Then, ρ(τ0)=d[η0,ˉη0]. Also, from (3.7), for any ϰ[0,1] we obtain

    ρ(τ)=d[η0+ττ0fϰ(σ,η(σ))dσ,ˉη0+ττ0fϰ(σ,ˉη(σ))dσ]  d[ττ0fϰ(σ,η(σ))dσ,ττ0fϰ(σ,ˉη(σ))dσ]+d[η0,ˉη0]. (5.2)

    Using (5.2) we get

    ρ(τ)ρ(τ0)+ττ0d[fϰ(σ,η(σ)),fϰ(σ,ˉη(σ))]dσρ(τ0)+ττ0g(σ,d[η(σ),ˉη(σ)])dσ=ρ(τ0)+ττ0g(σ,ρ(σ))dσ,τT. (5.3)

    Applying to (5.3) Theorem 1.6.1 from [38], we conclude that the estimate (5.1) is satisfied for any τT and ϰ[0,1].

    Remark 5.2. The estimate offered in Theorem 5.1 is based on the integral equation, and on the existence and uniqueness results established in Section 4. Such estimations are crucial in the investigation of the qualitative properties of the solutions when the Lyapunov method is applied. Hence, the proposed result can be developed by the use of the Lyapunov technique in the study of the stability, periodicity and almost periodicity behavior of the states of the regularized system (3.6).

    Condition 2 of Theorem 5.1 can be weakened while maintaining its statement.

    Theorem 5.3. Assume that Condition 1 of Theorem 5.1 holds, and that the following are true:

    1) There exists a family of functions gϰC(T×R+,R) such that

    limsup{[d[η+hfϰ(τ,η),ˉη+hfϰ(τ,ˉη)]]h1d[η,ˉη]:h0+}gϰ(τ,d[η,ˉη])

    for any ϰ[0,1] and (τ,η),(τ,ˉη)R+×EN.

    2) The maximal solution uϰ(τ,τ0,y0) of the scalar problem

    dy/dτ=gϰ(τ,y),y(τ0)=y00. (5.4)

    is defined on T.

    Then,

    d[η(τ),ˉη(τ)]¯u(τ,τ0,y0), τT,

    where η(τ) and ˉη(τ) are any two solutions of the problem (3.6) defined on T, corresponding to initial data (η0,ˉη0) such that d[η0,ˉη0]y0 and ¯u(τ,τ0,y0)=maxϰuϰ(τ,τ0,y0).

    Proof. Denote again ρ(t)=d[η(τ),ˉη(τ)]. Then, for the difference ρ(τ+h)ρ(τ), h>0, we have

    ρ(τ+h)ρ(τ)=d[η(τ+h),ˉη(τ+h)]d[η(τ),ˉη(τ)]
    d[η(τ+h),η(τ)+hfϰ(τ,η(τ))]+d[ˉη(τ)+hfϰ(τ,ˉη(τ)),ˉη(τ+h)]
    +d[hfϰ(τ,η(τ)),hfϰ(τ,ˉη(τ))]d[η(τ),ˉη(τ)],ϰ[0,1].

    From the above inequalities, we get

    D+ρ(τ)=limsup{[ρ(τ+h)ρ(τ)]h1:h0+}limsup{[d[η(τ)+hfϰ(τ,η(τ)),ˉη(τ)+hfϰ(τ,ˉη(τ))]]h1:h0+}d[η(τ),ˉη(τ)]+limh0+sup{[d[η(τ+h)η(τ)h,fϰ(τ,η(τ))]]}+limh0+sup{d[fϰ(τ,ˉη(τ)),ˉη(τ+h)ˉη(τ)h]}gϰ(τ,d[η,ˉη])=gϰ(τ,ρ(τ)),τTfor allϰ[0,1]. (5.5)

    The conclusion of Theorem 5.3 follows in the same way as in Theorem 5.1 by applying Theorem 1.6.1 from [38] to (5.5). Hence, for any ϰ[0,1] we get

    d[η(τ),ˉη(τ)]uϰ(τ,τ0,y0)

    and, therefore, d[η(τ),ˉη(τ)]¯u(τ,τ0,y0) for τT. The proof of Theorem 5.2 is complete.

    Theorems 5.1 and 5.3 offer also the opportunities for estimating the distance between an arbitrary solution η(τ) of the regularized problem (3.6) and the "steady state" θ0EN of (3.6).

    Corollary 5.4. Assume that Condition 1 of Theorem 5.1 holds, and that the family of functions gϰC(T×R+,R) is such that

    (a) d[fϰ(τ,η),θ0]gϰ(τ,d[η,θ0]) or

    (b) limsup{[d[η+hfϰ(τ,η),θ0]d[η,θ0]]h1:h0+}gϰ(τ,d[η,θ0]) for all τT, ϰ[0,1].

    Then d[η0,θ0]y0 implies

    d[η(τ),θ0]¯u(τ,τ0,y0),τT, (5.6)

    where ¯u(τ,τ0,y0)=maxϰuϰ(τ,τ0,y0), uϰ(τ,τ0,y0) is the maximal solution of the family of comparison problem

    dy/dτ=gϰ(τ,y),y(τ0)=y00,

    and η(τ) is an arbitrary solution of the problem (3.6) defined on T, corresponding to the initial value η0.

    Corollary 5.5. If in Corollary 5.4, gϰ(τ,d[η,θ0])=λ(τ)d[η,θ0] with λ(τ)>0 for τT, then the estimate (5.6) has the form

    d[η(τ),θ0]d[η0,θ0]exp(ττ0λ(σ)dσ),τT

    for any ϰ[0,1].

    Remark 5.6. All established results for the regularized system (3.6) can be applied to obtain corresponding results for the uncertain fuzzy problem (3.1). Thus, the proposed regularized scheme offers a new approach to study a class of fuzzy differential systems with uncertainties via systems of type (3.6), which significantly simplifies their analysis and is very appropriate for applied models of type (3.1). This will be demonstrated by the next example.

    Remark 5.7. The proposed regularized scheme and the corresponding approach can be extended to more general systems considering delay effects, impulsive effects and fractional-order dynamics.

    Example 5.8. We will apply Theorem 5.1 to estimate the distance between a solution of the problem (3.1) and the equilibrium state θ0. To this end, we consider a particular function g(τ,ζ) from Condition 2 of Theorem 5.1. We transform the fuzzy equation in (3.1) by using the regularized process to the form

    dξdτ=fϰ(τ,ξ)+g(τ,ξ,μ), (5.7)

    where g(τ,ξ,μ)=f(τ,ξ,μ)fϰ(τ,ξ) for all μS. Further we will suppose that fϰC(T×EN,EN) for all ϰ[0,1] and gC(T×EN×S,EN), T[τ0,α], g(τ,0,μ)0 for all ττ0.

    Let fϰ(τ,ξ) and g(τ,ξ,μ) be such that for all τT there exist continuous positive functions Ω(τ) and o(τ) satisfying the hypotheses

    1) d[fϰ(τ,ξ),θ0]Ω(τ)d[ξ,θ0] for all ϰ[0,1];

    2) d[g(τ,ξ,μ),θ0]o(τ)dq[ξ,θ0] for all μS;

    3) ψ(τ0,τ)=(q1)dq1[ξ0,θ0]ττ0o(σ)exp[(q1)στ0Ω(υ)dυ]dσ<1, q>1.

    For the family of equations (5.7), we assume that Hypotheses 13 are fulfilled for all τ,σ[τ0,α].

    Then, the deviations of any solution ξ(τ) of (5.7) from the state θ0EN are estimated as follows

    d[ξ(τ),θ0]d[ξ0,θ0]exp(ττ0Ω(σ)dσ)(1ψ(τ0,τ))1q1 (5.8)

    for all τ[τ0,α].

    From (5.7), we have

    ξ(τ)=ξ(τ0)+ττ0fϰ(σ,ξ(σ))dσ+ττ0g(σ,ξ(σ),μ)dσ. (5.9)

    Let z(t)=d[ξ(τ),θ0]. Then z(τ0)=d[ξ0,θ0], and

    d[ξ(τ),θ0]d[ξ0,θ0]+d[(ττ0fϰ(σ,ξ(σ))dσ+ττ0g(σ,ξ(σ),μ)dσ),θ0]d[ξ0,θ0]+ττ0d[fϰ(σ,ξ(σ)),θ0]dσ+ττ0d[g(σ,ξ(σ),μ),θ0]dσ. (5.10)

    In view of Hypotheses 1 and 2, the inequality (5.10) yields

    d[ξ(τ),θ0]d[ξ0,θ0]+ττ0(Ω(σ)d[ξ(σ),θ0]+o(σ)dq[ξ(σ),θ0])dσ

    or

    z(τ)z(τ0)+ττ0(Ω(σ)+o(σ)zq1(σ))z(σ)dσ, τ[τ0,α]. (5.11)

    Applying the estimation technique from [39] to inequality (5.11), we obtain the estimate

    zq1(τ)zq1(τ0)exp((q1)ττ0Ω(σ)dσ)1ψ(τ0,τ), τ[τ0,α]. (5.12)

    From (5.12) we obtain (5.8).

    If the Hypotheses 1 and 2 are satisfied and 0<q<1, then by applying Theorem 2.2 from [40] to the inequality (5.11), we get the estimate (5.8) in the form

    d[ξ(τ),θ0]d[ξ0,θ0]exp(1ψ(τ0,τ))1q1 (5.13)

    for all τ[τ0,α].

    In this paper we investigate some fundamental properties of fuzzy differential equations using a new approach. We introduce a regularization scheme for systems of fuzzy differential equations with uncertain parameters. Existence and uniqueness criteria for the regularized equations are established. Estimates about the distance between solutions of the regularized equations are also proposed. The proposed technique and the new results will allow us to consider the qualitative properties of the solutions such as stability, boundedness, periodicity, etc. The introduced approach can also be extended to study delayed systems and impulsive control systems via modifications of the Lyapunov theory [41,42,43,44]. The advantages of the proposed regularization procedure can be implemented in various fuzzy models such as fuzzy neural networks with uncertain parameters, fuzzy models in biology with uncertain parameters, fuzzy models in economics with uncertain parameters, and much more. Numerical applications of our finding in a way that is similar to [45] are also interesting and challenging further directions of research.

    The authors declare no conflict of interest.



    [1] J. Guo, M. Cui, C. Hou, G. Gou, Z. Li, G. Xiong, et al., Global-aware prototypical network for few-shot encrypted traffic classification, in 2022 IFIP Networking Conference (IFIP Networking), (2022), 1–9. https://doi.org/10.23919/IFIPNetworking55013.2022.9829771
    [2] S. Stryczek, M. Natkaniec, Internet threat detection in smart grids based on network traffic analysis using lstm, if, and svm, Energies, 16 (2023), 329. https://doi.org/10.3390/en16010329 doi: 10.3390/en16010329
    [3] H. Liu, B. Lang, Network traffic classification method supporting unknown protocol detection, in 2021 IEEE 46th Conference on Local Computer Networks (LCN), (2021), 311–314. https://doi.org/10.1109/LCN52139.2021.9525009
    [4] A. Barnawi, S. Gaba, A. Alphy, A. Jabbari, I. Budhiraja, V. Kumar, et al., A systematic analysis of deep learning methods and potential attacks in internet-of-things surfaces, Neural Comput. Appl., 2023 (2023), 1–16. https://doi.org/10.1007/s00521-023-08634-6 doi: 10.1007/s00521-023-08634-6
    [5] A. Yadav, S. Gaba, H. Khan, I. Budhiraja, A. Singh, K. K. Singh, Etma: Efficient transformer-based multilevel attention framework for multimodal fake news detection, IEEE Trans. Comput. Soc. Syst., 2023 (2023), forthcoming. https://doi.org/10.1109/TCSS.2023.3255242 doi: 10.1109/TCSS.2023.3255242
    [6] R. Moreira, L. F. Rodrigues, P. F. Rosa, R. L. Aguiar, F. de Oliveira Silva, Packet vision: a convolutional neural network approach for network traffic classification, in 2020 33rd SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI), (2020), 256–263. https://doi.org/10.1109/SIBGRAPI51738.2020.00042
    [7] K. Lin, X. Xu, Y. Jiang, A new semi-supervised approach for network encrypted traffic clustering and classification, in 2022 IEEE 25th International Conference on Computer Supported Cooperative Work in Design (CSCWD), (2022), 41–46. https://doi.org/10.1109/CSCWD54268.2022.9776310
    [8] J. Zhao, X. Liu, Q. Yan, B. Li, M. Shao, H. Peng, Multi-attributed heterogeneous graph convolutional network for bot detection, Inf. Sci., 537 (2020), 380–393. https://doi.org/10.1016/j.ins.2020.03.113 doi: 10.1016/j.ins.2020.03.113
    [9] P. Singh, G. Bathla, D. Panwar, A. Aggarwal, S. Gaba, Performance evaluation of genetic algorithm and flower pollination algorithm for scheduling tasks in cloud computing, in International Conference on Signal Processing and Integrated Networks, (2022), 139–154. https://doi.org/10.1007/978-981-99-1312-1_12
    [10] S. Gaba, I. Budhiraja, V. Kumar, S. Garg, G. Kaddoum, M. M. Hassan, A federated calibration scheme for convolutional neural networks: Models, applications and challenges, Comput. Commun., 192 (2022), 144–162. https://doi.org/10.1016/j.comcom.2022.05.035 doi: 10.1016/j.comcom.2022.05.035
    [11] A. Aggarwal, S. Gaba, J. Kumar, S. Nagpal, Blockchain and autonomous vehicles: Architecture, security and challenges, in 2022 Fifth International Conference on Computational Intelligence and Communication Technologies (CCICT), IEEE, (2022), 332–338. https://doi.org/10.1109/CCiCT56684.2022.00067
    [12] Y. Wang, X. Yun, Y. Zhang, C. Zhao, X. Liu, A multi-scale feature attention approach to network traffic classification and its model explanation, IEEE Trans. Network Serv. Manage., 19 (2022), 875–889. https://doi.org/10.1109/TNSM.2022.3149933 doi: 10.1109/TNSM.2022.3149933
    [13] J. Zhao, M. Shao, H. Wang, X. Yu, B. Li, X. Liu, Cyber threat prediction using dynamic heterogeneous graph learning, Knowl. Based Syst., 240 (2022), 108086. https://doi.org/10.1016/j.knosys.2021.108086 doi: 10.1016/j.knosys.2021.108086
    [14] Q. Ma, W. Huang, Y. Jin, J. Mao, Encrypted traffic classification based on traffic reconstruction, in 2021 4th International Conference on Artificial Intelligence and Big Data (ICAIBD), IEEE, (2021), 572–576. https://doi.org/10.1109/ICAIBD51990.2021.9459072
    [15] Y. Zeng, Z. Qi, W. Chen, Y. Huang, Test: an end-to-end network traffic classification system with spatio-temporal features extraction, in 2019 IEEE International Conference on Smart Cloud (SmartCloud), IEEE, (2019), 131–136. https://doi.org/10.1109/SmartCloud.2019.00032
    [16] A. Aggarwal, S. Gaba, S. Nagpal, A. Arya, A deep analysis on the role of deep learning models using generative adversarial networks, in Blockchain and Deep Learning: Future Trends and Enabling Technologies, Springer, (2022), 179–197. https://doi.org/10.1007/978-3-030-95419-2_9
    [17] S. Nagpal, A. Aggarwal, S. Gaba, Privacy and security issues in vehicular ad hoc networks with preventive mechanisms, in Proceedings of International Conference on Intelligent Cyber-Physical Systems: ICPS 2021, Springer, (2022), 317–329. https://doi.org/10.1007/978-981-16-7136-4_24
    [18] G. Aceto, D. Ciuonzo, A. Montieri, A. Pescapé, Mobile encrypted traffic classification using deep learning: Experimental evaluation, lessons learned, and challenges, IEEE Trans. Network Serv. Manage., 16 (2019), 445–458. https://doi.org/10.1109/TNSM.2019.2899085 doi: 10.1109/TNSM.2019.2899085
    [19] M. Lotfollahi, M. J. Siavoshani, R. S. Hossein Zade, M. Saberian, Deep packet: A novel approach for encrypted traffic classification using deep learning, Soft Comput., 24 (2020), 1999–2012. https://doi.org/10.1007/s00500-019-04030-2 doi: 10.1007/s00500-019-04030-2
    [20] G. Aceto, D. Ciuonzo, A. Montieri, A. Pescapé, MIMETIC: Mobile encrypted traffic classification using multimodal deep learning, Comput. Networks, 165 (2019), 106944. https://doi.org/10.1016/j.comnet.2019.106944 doi: 10.1016/j.comnet.2019.106944
    [21] M. Lopez-Martin, B. Carro, A. Sanchez-Esguevillas, J. Lloret, Network traffic classifier with convolutional and recurrent neural networks for Internet of Things, IEEE Access, 5 (2017), 18042–18050. https://doi.org/10.1109/ACCESS.2017.2747560 doi: 10.1109/ACCESS.2017.2747560
    [22] J. Li, V. S. Sheng, Z. Shu, Y. Cheng, Y. Jin, Y. F. Yan, Learning from the crowd with neural network, in 2015 IEEE 14th International Conference on Machine Learning and Applications (ICMLA), (2015), 693–698. https://doi.org/10.1109/ICMLA.2015.14
    [23] X. Y. Zhang, G. S. Xie, C. L. Liu, Y. Bengio, End-to-end online writer identification with recurrent neural network, IEEE Trans. Human Mach. Syst., 47 (2016), 285–292. https://doi.org/10.1109/THMS.2016.2634921 doi: 10.1109/THMS.2016.2634921
    [24] X. Shi, H. Qi, Y. Shen, G. Wu, B. Yin, A spatial–temporal attention approach for traffic prediction, IEEE Trans. Intell. Transp. Syst., 22 (2020), 4909–4918. https://doi.org/10.1109/TITS.2020.2983651 doi: 10.1109/TITS.2020.2983651
    [25] Y. Saadna, A. Behloul, An overview of traffic sign detection and classification methods, Int. J. Multimedia Inf. Retr., 6 (2017), 193–210. https://doi.org/10.1007/s13735-017-0129-8 doi: 10.1007/s13735-017-0129-8
    [26] D. Kaur, A. Anwar, I. Kamwa, S. Islam, S. M. Muyeen, N. Hosseinzadeh, A Bayesian deep learning approach with convolutional feature engineering to discriminate cyber-physical intrusions in smart grid systems, IEEE Access, 11 (2023), 18910–18920. https://doi.org/10.1109/ACCESS.2023.3247947 doi: 10.1109/ACCESS.2023.3247947
    [27] A. Aldweesh, A. Derhab, A. Z. Emam, Deep learning approaches for anomaly-based intrusion detection systems: A survey, taxonomy, and open issues, Knowl. Based Syst., 189 (2020), 105124. https://doi.org/10.1016/j.knosys.2019.105124 doi: 10.1016/j.knosys.2019.105124
    [28] J. Bhardwaj, J. P. Krishnan, D. F. L. Marin, B. Beferull-Lozano, L. R. Cenkeramaddi, C. Harman, Cyber-physical systems for smart water networks: A review, IEEE Sens. J., 21 (2021), 26447–26469. https://doi.org/10.1109/JSEN.2021.3121506 doi: 10.1109/JSEN.2021.3121506
    [29] M. S. Akhtar, T. Feng, Detection of malware by deep learning as CNN-LSTM machine learning techniques in real time, Symmetry, 14 (2022), 2308. https://doi.org/10.3390/sym14112308 doi: 10.3390/sym14112308
    [30] D. D. Godsey, Y. H. Hu, M. A. Hoppa, A Multi-layered Approach to Fake News Identification, Measurement and Mitigation, in Advances in Information and Communication: Proceedings of the 2021 Future of Information and Communication Conference (FICC), (2021), 624–642. https://doi.org/10.1007/978-3-030-73100-7_45
    [31] Y. Jang, N. Kim, B. D. Lee, Traffic classification using distributions of latent space in software-defined networks: An experimental evaluation, Eng. Appl. Artif. Intell., 119 (2023), 105736. https://doi.org/10.1016/j.engappai.2022.105736 doi: 10.1016/j.engappai.2022.105736
    [32] A. V. Jain, Network traffic identification with convolutional neural networks, in 2018 IEEE 16th Intl Conf on Dependable, Autonomic and Secure Computing, 16th Intl Conf on Pervasive Intelligence and Computing, 4th Intl Conf on Big Data Intelligence and Computing and Cyber Science and Technology Congress (DASC/PiCom/DataCom/CyberSciTech), IEEE, (2018), 1001–1007.
    [33] S. Dong, Multi class svm algorithm with active learning for network traffic classification, Expert Syst. Appl., 176 (2021), 114885. https://doi.org/10.1016/j.eswa.2021.114885 doi: 10.1016/j.eswa.2021.114885
    [34] Y. Guo, G. Xiong, Z. Li, J. Shi, M. Cui, G. Gou, Combating imbalance in network traffic classification using gan based oversampling, in 2021 IFIP Networking Conference (IFIP Networking), IEEE, (2021), 1–9. https://doi.org/10.23919/IFIPNetworking52078.2021.9472777
    [35] F. Al-Obaidy, S. Momtahen, M. F. Hossain, F. Mohammadi, Encrypted traffic classification based ml for identifying different social media applications, in 2019 IEEE Canadian Conference of Electrical and Computer Engineering (CCECE), IEEE, (2019), 1–5. https://doi.org/10.1109/CCECE.2019.8861934
    [36] X. Ren, H. Gu, W. Wei, Tree-rnn: Tree structural recurrent neural network for network traffic classification, Expert Syst. Appl., 167 (2021), 114363. https://doi.org/10.1016/j.eswa.2020.114363 doi: 10.1016/j.eswa.2020.114363
    [37] W. Liu, C. Zhu, Z. Ding, H. Zhang, Q. Liu, Multiclass imbalanced and concept drift network traffic classification framework based on online active learning, Eng. Appl. Artif. Intell., 117 (2023), 105607. https://doi.org/10.1016/j.engappai.2022.105607 doi: 10.1016/j.engappai.2022.105607
    [38] Y. Pan, X. Zhang, H. Jiang, C. Li, A network traffic classification method based on graph convolution and lstm, IEEE Access, 9 (2021), 158261–158272. https://doi.org/10.1109/ACCESS.2021.3128181 doi: 10.1109/ACCESS.2021.3128181
    [39] C. Gijón, M. Toril, M. Solera, S. Luna-Ramírez, L. R. Jimenez, Encrypted traffic classification based on unsupervised learning in cellular radio access networks, IEEE Access, 8 (2020), 167252–167263. https://doi.org/10.1109/ACCESS.2020.3022980 doi: 10.1109/ACCESS.2020.3022980
    [40] X. Jing, J. Zhao, Z. Yan, W. Pedrycz, X. Li, Granular classifier: Building traffic granules for encrypted traffic classification based on granular computing, Dig. Commun. Networks, 2022 (2022), forthcoming. https://doi.org/10.1016/j.dcan.2022.12.017 doi: 10.1016/j.dcan.2022.12.017
    [41] S. Ahn, J. Kim, S. Y. Park, S. Cho, Explaining deep learning-based traffic classification using a genetic algorithm, IEEE Access, 9 (2020), 4738–4751. https://doi.org/10.1109/ACCESS.2020.3048348 doi: 10.1109/ACCESS.2020.3048348
    [42] J. Zhang, J. Zhou, N. Zhou, Network traffic classification method based on subspace triple attention mechanism, in 2022 3rd International Conference on Information Science, Parallel and Distributed Systems (ISPDS), IEEE, (2022), 312–316. https://doi.org/10.1109/ISPDS56360.2022.9874195
    [43] A. S. Iliyasu, H. Deng, Semi-supervised encrypted traffic classification with deep convolutional generative adversarial networks, IEEE Access, 8 (2019), 118–126. https://doi.org/10.1109/ACCESS.2019.2962106 doi: 10.1109/ACCESS.2019.2962106
    [44] L. K. Ramasamy, F. Khan, M. Shah, B. V. V. S. Prasad, C. Iwendi, C. Biamba, Secure smart wearable computing through artificial intelligence-enabled internet of things and cyber-physical systems for health monitoring, Sensors, 22 (2022), 1076. https://doi.org/10.3390/s22031076 doi: 10.3390/s22031076
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