Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Raheleh Jafari; Wen Yu; Xiaoou Li; Sina Razvarz

doi:10.2991/ijcis.10.1.81

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Volume 10, Issue 1, 2017, Pages 1226 - 1237

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Authors

Raheleh Jafari¹^,jafari3339@yahoo.com, Wen Yu¹^{, *}^,yuw@ctrl.cinvestav.mx, Xiaoou Li²^,lixo@cs.cinvestav.mx, Sina Razvarz¹^,srazvarz@yahoo.com

Departamento de Control Automático, CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico

Departamento de Computación, CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico

^*The corresponding author is Wen Yu, tel.: +52-55-57473734, email: yuw@ctrl.cinvestav.mx.

Corresponding Author

Wen Yuyuw@ctrl.cinvestav.mx

Received 6 November 2016, Accepted 4 September 2017, Available Online 19 September 2017.

DOI: 10.2991/ijcis.10.1.81 How to use a DOI?
Keywords: Fuzzy differential equations; Bernstein neural networks; Z- numbers; Uncertain nonlinear systems
Abstract: The uncertain nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs) by incorporating the fuzzy set theory. The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs.

In this paper, the solutions of FDEs are approximated by two types of Bernstein neural networks. Here, the uncertainties are in the sense of Z-numbers. Initially the FDE is transformed into four ordinary differential equations (ODEs) with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z-number variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs based on Z-numbers.
Copyright: © 2017, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Since the uncertainty in parameters can be transformed into fuzzy set theory [1] , fuzzy set and fuzzy system theory are good tools to deal with uncertainty systems. Fuzzy models are applied for a large class of uncertainty nonlinear systems, for example Takagi-Sugeno fuzzy model [2] . When the parameter of an equation are changeable in the manner of fuzzy set, this equation becomes a fuzzy equation [3] . When the parameters or the states of the differential equations are uncertain, they can be modeled with FDE.

Many FDEs use fuzzy numbers as the coefficients of the differential equations to describe the uncertainties[4] . The applications of these FDEs are connection with nonlinear modeling and control [5–8]. Another type of FED uses fuzzy variables to express the uncertainties. The study on the solutions of FDEs are applied into chaotic analysis, quantum system and many engineering problems, such as civil engineering and modeling actuators. The basic idea of fuzzy derivative was first introduced in [9] . Then it is extended in [10] . The linear first-order equation is the most generalized FDE. By generalizing the differentiability, [11] gives an analytical solution. In [12] , the first order FDE with periodic boundary conditions is analyzed. Then higher order linear FDE are studied. In [13] , the analytical solutions of second order FDE are obtained. The analytical solutions of third order linear FDE are found in [14] .

Too much complexity is involved in solving nonlinear FDE. By interval-valued method, [15] examines the basis solutions nonlinear FDEs with generalized differentiability. [16] uses periodic boundary and Hukuhara differentiability to the impulsive FDE. [17] suggests some suitable criterion to fuzzify the crisp solutions. [18] uses two-point fuzzy boundary value for FDE. However, all of above analytical methods for the solutions of FDEs are very difficult, especially for nonlinear FDEs.

Numerical solutions of FDEs have been discussed by many scientists recently. The numerical solutions of first-order FDE is proposed in [19] with an iterative technique. [20] uses Laplace transform for second-order FDE. By extending classical fuzzy set theory, [21] obtains numerical solution of an FDE. The predictor-corrector approach is applied in [22] . Euler numerical technique is used in [23] to solve FDE. Some other numerical techniques, such as Nystrom approach [24] , Taylor method [25] and Runge-Kutta approach [26] can also be applied to solve FDEs. However the approximation accuracy of these numerical calculations are normally less.

The solution of FDE is uniformly continuous and inside compact sets [27] . Neural networks can give a good estimation for the solutions of FDEs. [28] shows that the solution of ODE can be approximated by neural network. [29] discusses differences between the exact solution and approximation solutions of ODEs. [30] applies dynamics neural networks to approximate first-order ODE. There are few works on FDE. [31] suggests a static neural network to solve FDE. Since the structure of the neural network is not suitable for FDE, the approximation accuracy is poor.

The decisions are carried out based on knowledge. In order to make the decision fruitful, the knowledge acquired must be credible. Z-numbers connect to the reliability of knowledge [32] . Many fields related to the analysis of the decisions are actually use the ideas of Z-numbers. Z -numbers are much less complex to calculate compared with nonlinear system modeling methods. The Z-number is abundantly adequate number compared with the fuzzy number. Although Z-numbers are implemented in many literatures, from theoretical point of view this approach is not certified completely.

The Z-number is a novel idea that is subjected to a higher potential to illustrate the information of the human being and to use in information processing [32] . Z-numbers can be regarded as to answer questions and carry out the decisions [33] . There are few structure based on the theoretical concept of Z-numbers [34] . [35] proposes a theorem to transfer the Z-numbers to the usual fuzzy sets.

In this paper, a new model named Bernstein neural network is used, which has good properties of Bernstein polynomial for FDE based on Z-number. The Bernstein polynomial has good uniform approximation ability for continuous functions [36] . Also it has innumerable drawing properties, homogeneous shape-sustaining approximation, bona fide estimation and low boundary bias. A very important property of the Bernstein polynomial is that it generates a smooth estimation for equal distance knots [37] . This property is suitable for FDE approximation. For more details regarding Bernstein neural networks, refer to [38, 39].

Two types of neural networks are used namely static and dynamic models, to approximate the solutions of FDEs based on z -numbers. These numerical methods use generalized differentiability of FDEs. The solutions of FDE is substituted into four ODEs. Then the corresponding Bernstein neural networks are applied. Finally, some real examples are used to show the effectiveness of the proposed approximation methods with the Bernstein neural networks.

2. Fuzzy differential equation for uncertain nonlinear system modeling

Consider the following controlled unknown nonlinear system

(1)x˙=f1(x1,u,t)

where f₁ (x₁, u, t) is unknown vector function, x₁ ∈ ℜ ⁿ is an internal state vector and u ∈ ℜ ^m is the input vector.

In this paper, the following differential equation (FDE) is used to model the uncertain nonlinear system (1) ,

(2)ddtx=f(t,x)

where x ∈ ℜⁿ is the Z-number variable, which corresponds to the state x₁ in (1) , f(t, x) is a Z-number vector function, which relates to f₁(x₁, u), ddtx is the derivative associated to the Z-number variable. Here the uncertainties of the nonlinear system (1) are in the sense of Z-numbers.

In order to use FDE based on Z- numbers, initially some concepts of fuzzy variables and Z-numbers.

Definition 1

(fuzzy variable) If x is:

1)
normal, there exists ζ₀ ∈ R in such a manner x(ζ₀) = 1,
2)
convex, x(λζ + (1 – λ)ζ) ≥ min{x(ζ), x(ξ)}, ∀ ζ, ξ ∈ R, ∀ λ ∈ [0,1],
3)
upper semi-continuous on R, x(ζ) ≤ x(ζ₀) + ε, ∀ ζ∈ N(ζ₀), ∀ζ₀ ∈ R, ∀ ε > 0, N(ζ₀) is a neighborhood,
4)
x⁺ = {ζ∈ ℝ, x(ζ) ∈ E|ℝ → [0,1]} is compact, then x is a fuzzy variable.

The fuzzy variable x can be also represented as [40]

(3)x=A(x_,x_)

where x is the lower-bound variable, x¯ is the upper-bound variable and A is a continuous function.

Definition 2

(Z- number) A Z-number has two components Z = [x(ζ), p]. The primary component x(ζ) is termed as a restriction on a real-valued uncertain variable ζ. The secondary component p is a measure of reliability of x. p can be reliability, strength of belief, probability or possibility. When x(ζ) is a fuzzy number and p is the probability distribution of ζ, the Z-number is defined as Z⁺-number. When both x(ζ) and p are fuzzy numbers, the Z-number is defined as Z⁻ -number.

The Z⁺ -number carries more information than the Z⁻ -number. In this paper, the definition of Z⁺ -number is used, i.e., Z = [x, p] x is a fuzzy number and p is a probability distribution.

We use so called membership functions to express the fuzzy number. The most popular membership functions are the triangular function

(4)μx(ζ)=F(a,b,c)={ζ−ab−a a≤ζ≤bc−ζc−b b≤ζ≤cotherwise μx=0

and trapezoidal function

(5)μx(ζ)=F(a,b,c,d)={ζ−ab−aa≤ζ≤bd−ζd−cc≤ζ≤d1b≤ζ≤cotherwise μx=0

The probability measure is expressed as

(6)P=∫Rμx(ζ)p(ζ)dζ

where p is the probability density of ζ and R is the restriction on p. For discrete Z-numbers

(7)P(x)=∑i=1nμx(ζi)p(ζi)

Definition 3

(α - level of fuzzy number) The fuzzy number x in association to the α -level is illustrated as

(8)[x]α={a∈R : x(a)≥α}

where 0 ≤ α ≤ 1, x ∈ E.

Therefore [x]⁰ = x⁺ = {ζ∈ R, x(ζ) > 0}. Since α ∈ [0,1], [x]^α is a bounded mentioned as x_α≤[x]α≤x¯α . The α-level of x in midst of x^α and x¯α is explained as

(9)[x]α=A(x_α,x_α)

x^α and x¯α signify the function of α, and x^α = d_M(α), x¯α=dU(α) , α ∈ [0,1].

Definition 4

(α - level of Z-numbers) The α -level of the Z-number Z = (x, P) is demonstrated as

(10)[Z]α=([x]α,[p]α)

where 0 ≤ α ≤ 1. [p]^α is calculated by the Nguyen’s theorem

[p]α=p([x]α)=p([x_α,x_α])=⌊P_α,P_α⌋

where p([x]^α) = {p(ζ)|ζ∈[x]^α}. So [Z]^α can be expressed as the form α -level of a fuzzy number

(11)[Z]α=(Z_α,Z_α)=((x_α,P_α),(x_α,P_α))

where P_α=x_αp(ζi_α) , P¯α=x¯αp(ζi¯α) , [ζi]α=(ζi_α,ζi¯α) .

Similar with the fuzzy numbers [5] , the Z-numbers are also incorporated with three primary operations: ⊕, ⊝and ⊙. These operations are exhibited by: sum, subtract, multiply, and division. The operations in this paper are different definitions with [1] . The α-level of Z-numbers is applied to simplify the operations.

Let us consider Z₁ = (x₁, p₁) and Z₂ = (x₂, p₂) be two discrete Z-numbers illustrating the uncertain variables ζ₁ and ζ₂, ∑k=1np1(ζ1k)=1 , ∑k=1np2(ζ2k)=1 . The operations are defined

Z12=Z1*Z2=(x1*x2,p1*p2)

where *∈ {⊕, ⊝, ⊙}.

The operations for the fuzzy numbers are defined as [5]

(12)[x1⊕x2]α=[x1_α+x2_α,x1 α+x2 α][x1⊖x2]α=[x1_α−x2_α,x1 α−x2 α][x1⊙x2]α=[x1_αx2_α+x1_αx2_α−x1_αx2_α,x1 αx2 α+x1 αx2 α−x1 αx2 α]

For all p₁ * p₂ operations, we use convolutions for the discrete probability distributions

p1*p2=∑ip1(ζ1,i)p2(ζ2,(n−i))=p12(ζ)

The above definitions satisfy the Hukuhara difference [41] ,

Z1⊖HZ2= Z12 Z1=Z2⊕Z12

Here if Z₁ ⊝_H Z₂ exists, the α -level is

[Z1⊖HZ2]α=[Z_1α−Z_2α,Z_1α−Z_2α]

Obviously, Z₁ ⊝_H Z₁ = 0, Z₁ ⊝ Z₁ ≠ 0

Also the above definitions satisfy the generalized Hukuhara difference [42]

(13)Z1⊖gHZ2=Z12⇔{1)Z1=Z2⊕Z122)Z2=Z1⊕(−1)Ζ12

It is convenient to display that 1) and 2) in combination are genuine if and only if Z₁₂ is a crisp number. With respect to α -level what we got are [Z1⊖gHZ2]α=[min{Z_1α−Z_2α,Z¯1α−Z¯2α},max{Z_1α−Z_2α,Z¯1α−Z¯2α}] and If Z₁ ⊝_gH Z₂ and Z₁ ⊝_H Z₂ subsist, Z₁ ⊝_H Z₂ = Z₁ ⊝_gH Z₂. The conditions for the existence of Z₁₂ = Z₁ ⊝_gH Z₂ ∈ E are

(14)1){Z_12α=Z_1α−Z_2α and Z_12α=Z_1α−Z_2αwith Z_12α increasing, Z_12α decreasing,Z_12α≤Z_12α2){Z_12α=Z_1α−Z_2α andZ_12α=Z_1α−Z_2αwith Z_12α increasing, Z 12α decreasing, Z_12α≤Z_12α

where ∀ α ∈ [0,1]

If x is a triangular function, the absolute value of the Z-number Z = (x, p) is

(15)|Z(ζ)|=(|a1|+|b1|+|c1|,p(|a2|+|b2|+|c2|))

If x₁ and x₂ are triangular functions, the supremum metric for Z-numbers Z₁ = (x₁, p₁) and Z₂ =(x₂, p₂) is given as

D(Z1,Z2) =d(x1,x2) +d(p1,p2)

in this case d(.,.)is the supremum metrics considering fuzzy sets [5] . D(Z₁, Z₂) is incorporated with the following possessions:

D(Z1+Z,Z2+Z) =D(Z1,Z2)D(Z2,Z1)=D(Z1,Z2)D(kZ1,kZ2)=|k| D(Z1,Z2)D(Z1,Z2)≤D(Z1, Z)+D(Z,Z2)

where k ∈ R, Z = (x, p) is Z-number and x is triangle function, for proof refer to [43].

Definition 5

(α-level of Z-number valued function) Let Z˜ denotes the space of Z-numbers. The α -level of Z-number valued function F:[0,a]→Z˜ is

F(x,α)=[F_(x,α),F_(x,α)]

Where ∈ Z˜ , for each α ∈ [0,1].

With the definition of Generalized Hukuhara difference, the gH-derivative of F at x₀ is expressed as

(16)ddtF(x0)=limh→01h[F(x0+h)⊖gHF(x0)]

In (16), F(x₀+h) and F(x₀) exhibits similar style with Z₁ and Z₂ respectively included in (13).

If we apply the α – level (10) to f(t, x) in (2), then we obtain two Z-number valued functions: f_[t,x_(ζ,α),x¯(ζ,a)] and f¯[t,x_(ζ,α),x¯(ζ,a)] .

The fuzzy differential equation (2) can be equivalent to the following four ODE

(17)1){ddtx_=f_[t,x_(ζ,α),x_(ζ,α)]ddtx_=f_[t,x_(ζ,α),x_(ζ,α)]2){ddtx_=f_[t,x_(ζ,α),x_(ζ,α)]ddtx_=f_[t,x_(ζ,α),x_(ζ,α)]

The fuzzy model of (1) can be regarded as four ordinary differential equations (17).

In this paper, the FDE (2) is used to model the uncertain nonlinear system (1), such that the output of the plant x can follow the plant output x₁,

(18)minf‖x−x1‖

This modeling object can be considered as: finding f¯ and f in the fuzzy equations of (17) or finding the solutions of these models. It is impossible to obtain analytical solutions [44].

In fact, the nonlinear system can be modeled by the neural network directly. However, this data-driven black box identification method does not use the model information. While the FDE use the model information of the nonlinear system, such as the brief form of the differential equation.

The following theorems give theory support for nonlinear system modeling via FDEs based on Z-numbers.

Theorem 1

If the Z-number function f and its derivative ∂f∂x are on the rectangle [−p, p] × [−q, q], here p, q∈Z˜ , Z˜ is the space of Z-numbers, then there exists an unique Z-number solution for the following FDE based on Z-numbers

(19)ddtx=f(t,x), x(t0)=x0

for all t ∈ (–b, b), b ≤ p

Proof.

We utilize Picard’s iteration technique [45] to develop a sequence of Z-number functions φ_n(t) as

ϕn+1(t)=ϕ0⊕∫0tf(s,ϕn(s))ds=ϕ0⊖H(−1)∫0tf(s,ϕn(s))ds

We first validate that φ_n(t) is continuous and exists for all n. Obviously, if φ_n(t) exists then φ_n₊₁(t) also exists as

ϕn+1(t)=ϕ0⊕∫0tf(s,ϕn(s))ds= ϕ0⊖H(−1)∫0tf(s,ϕn(s))ds

Since f is continuous, so there exists N ∈ E such that |f(t, x) |≤ N for all t ∈ [–p, p], as well as all x ∈ [–q, q]. If we set t ∈ [–b, b] for b ≤ min (q/N, p) then it is possible

||ϕn+1⊖ϕ0|| = ‖∫0tf(s,ϕn(s))ds‖≤N|t| ≤Nb≤q

This validates that φ_n₊₁(t) acquires values in [–q, q].

Because

ϕn(t)=∑k=1n(ϕk(t)⊖ϕk−1(t))

for any γ < 1, we select t ∈ [–b, b] such that |φ_k(t) ⊝φ_k₋₁ (t)| ≤ γ^k for all k. This signifies that there exists γ < 1[46]

|ϕk(t)⊖ϕk−1(t)|≤γk|

From the mean value theorem [47],

ϕk(t)⊖ϕk−1(t)=∫0t[f(s,ϕk−1(s))⊖f(s,ϕk−2(s))]ds

Applying the mean value theorem into the Z-number function h(x) = f (s, x) in the two points φ_k−1(s) and, φ_k−2(s),

h(ϕk−1(s))⊖h(ϕk−2(s)) =h′(ψk(s)) (ϕk−1(s))⊖ϕk−2(s)))

Taking into consideration h′(x)=∂f∂x , we obtain

(20)ϕk(t)ϕk−1(t)=∫0t∂f∂x(s,ψk(s))(ϕk−1(s)⊖ϕk−2(s))ds

Because |φ_k₋₁(s) − φ_k_–2(s)|≤ γ^k⁻¹ for s ≤ t and b < γ / N, by substituting the above relation in (20) and utilizing the boundess of ∂f∂x ,

|ϕk(t))⊖ϕk−1(t)|≤∫0tNγk−1ds=Ntγk−1≤Nbγk−1

In order to validate that x is continuous, it is necessary to show that for any given ε > 0 there exists δ > 0 in such a manner that |t₂ − t₁|< δ implies |φ(t₂) – φ(t₁)|<ε. At par with the notation convenience, we suppose that t₁ < t₂. It follows that

ϕ(t2)−ϕ(t1)=limn→∞ϕn(t2)−limn→∞ϕn(t1)=limn→∞(ϕn(t2)−ϕn(t1))=limn→∞∫t1t2f(s,ϕn(s))ds

There exists N in such a manner that |f(s, x) |≤ N. Hence

|ϕ(t2)−ϕ(t1)|≤∫t1t2Nds=N|t2−t1|≤Nδ

henceforth by selecting δ < ε/N it is observed that |φ(t₂) − φ(t₁)|< ε .So lim_n_→∞ φ_n(t) exists for all t.

Now we demonstrate that lim_n_→∞ φ_n(t) is continuous.

Since

ϕ(t)=limn→∞ϕn(t)=limn→∞∫0tf(s,ϕn−1(s))ds=∫0tlimn→∞f(s,ϕn−1(s))ds=∫0tf(s,limn→∞ϕn−1(s))ds

where the last step (moving the limit inside the function) is at par with the concept that f is continuous in each variable. Hence it is clear that

ϕ(t)=∫0tf(s,ϕ(s))ds

because all functions are continuous,

ddtϕ=f(s,ϕ(t))

If there exists another solution ϕ(t),

ϕ(t)−φ(t)=∫0t(f(s,ϕ(t))−f(s,φ(t)))ds

Since the two functions are different, there exists ε >0, |φ(t) − ϕ(t) |> ε . We define

m=max0≤t≤b|ϕ(t)−φ(t)|

N is the bound for ∂f∂x . Utilizing the mean value theorem,

|ϕ(t)−φ(t)|≤∫0tN|ϕ(t)−φ(t)|ds≤N|t|m≤Nbm

If we select b < ε/2mN, it signifies that for all t < b, |φ(t) − ϕ(t) |< ε/2, that contracts the fact that the least difference is ∊. So there exists an unique Z-number solution.

Theorem 2

Assume the following FDE based on Z-numbers

(21)ddtx=f(t,x)

here f∈J¯ab, J¯ab is the set of linear strongly bounded operators, for every operators f there exists a function such that | f (t, x) | ≤ τ (t)‖ν‖_G, t ∈ [a, b] and and there exist f₀, f₁ ∈ φ_ab, φ_ab is a set of linear operators f∈J¯ab from the set to the set such that

(22)|f_(t,x_,x_)+f_1(t,x_,x_)|≤f_0(t,|x_|,|x_|), t∈[a,b]|f_(t,x_,x_)+f_1(t,x_,x_)|≤f_0(t,|x_|,|x_|), t∈[a,b]

then (21) has an unique solution.

Proof.

If x is a Z-number solution of (21) and −12f1∈Jab(a) ,

(23)ddtβ=−12f1(t,β)⊕f0(t,|x|)⊕12f1(t,|x|)

contains a unique Z-number solution β. Moreover as f₀, f₁ ∈ φ_ab

(24)β_(t)≥0, t∈[a,b]β_(t)≥0, t∈[a,b]

According to (22) and the condition f₁ ∈ φ_ab, from (23) we have

ddtβ_≥−12f_1(t,β_,β_)+f_(t,x_,x_)+12f_1(t,x_,x_)ddtβ_≥−12f_1(t,β_,β_)+f_(t,x_,x_)+12f_1(t,x_,x_)

thus t ∈ [a,b]

ddt(−β_)≤−12f_1(t,−β_,−β_)+k_(t,x_,x_)+12k_1(t,x_,x_)ddt(−β_)≤−12f_1(t,−β_,−β_)+f_(t,x_,x_)+12f_1(t,x_,x_)

The last two inequalities is on account of the presumption −12f1∈Jab(a)

(25)|x_(t)|≤β_(t) t∈[a,b]|x_(t)|≤β_(t) t∈[a,b]

According to the (25) and the conditions f₀, f₁ ∈ φ_ab, (23) results in

ddtβ_≤f_0(t,β_,β_), t∈[a,b]ddtβ_≤f0(t,β_,β_), t∈[a,b]

As f₀ ∈ J_ab(a), the last inequality with β (a) = 0 yields β(t) ≤ 0 and β¯(t)≤0 for t ∈ [a,b]. (24) implies β ≡ 0. Thus based on (25) we have x ≡ 0.

3. Solving fuzzy differential equation with neural networks

In general, it is difficult to solve the four equations (17) or (2). In this paper, a special neural network named Bernstein neural network is used to approximate the solutions of the FDE (2).

The Bernstein neural network use the following Bernstein polynomial,

(26)B(x1,x2)=∑i=0N∑j=0M(Ni)(Mj)Wi,jx1i(T−x1i)N−ix2j(1−x2j)M−j

where (Ni)=N!i!(N−i)! , (Mj)=M!j!(M−j)! , W_i,j is the Z-number coefficient.

This two variable polynomial can be regarded as a neural network, which has two inputs x₁_i and x₂_j and one output y,

(27)y=∑i=0N∑j=0MλiγjWi,jx1i(T−x1i)N−ix2j(1−x2j)M−j

where λi=(Ni) , γj=(Mj) .

Because the Bernstein neural network (27) has similar forms as (17), the Bernstein neural network (27) is used to approximate the solutions of four ODEs in (17), see Figure 1.

If x₁ and x₂ in (26) are defined as: x₁ is time interval t, x₂ is the α -level, the solution of (2) in the form of the Bernstein neural network is

(28) xm(t,α)=xm(0,α)⊕t∑i=0N∑j=0MλiγjWi,jti(T−ti)N−iαj(1−αj)M−j

where x_m(0, α) is the initial condition of the solution based on Z-number.

So the derivative of (27) is

(29)1){ddtx_m=C1+C2ddtx_m=D1+D22){ddtx_m=C1+C2ddtx_m=D1+D2

where t ∈ [0, T], α ∈ [0,1], t_k = kh, h=Tk , k = 1,…, N, α_j = jh₁, h1=1M , j = 1,…, M,

C1=∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−jD1=∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−jC2=tk∑i=0N∑j=0MλiγjW_i,j[iti−1,j(T−ti)N−i−(N−i)ti,j(T−ti)N−j−1]αji(1−αj)M−jD2=tk∑i=0N∑j=0MλiγjW_i,j[iti−1,j(T−ti)N−i−(N−i)ti,j(T−ti)N−i−1]αji(1−αj)M−j

The above equations can be regarded as the neural network form, see Figure 2.

•
input unit:
o11=t,o21=α
•
the first hidden units:
o1,i2=fi1(o11),o2,i2=fi2(o11)o3,j2=gj1(o21),o4,j2=gj2(o21)
•
the second hidden units:
o1,i3=o1,i2(o2,i2),o2,j3=o3,j2(o4,j2)
•
the third hidden units:
o1,i4=λio1,i3,o2,i′4=γjo2,j3
•
the forth hidden units:
oi,j5=o1,i4o2,j4
•
output unit:
N(t,α)=∑i=0N∑j=0M(ai,joi,j5)
where fi1=ti , fi2=(T−t)N−i , λi=1TN(Ni) , gj1=αj , gj2=(1−α)M−j , γj=(Mj) .

The training errors between (29) and (17) are defined as

(30)1){e_1=C1+C2−f_e1 =D1+D2−f_2){e_2=C1+C2−f_e2 =D1+D2−f_

The standard back-propagation learning algorithm is utilized to update the weights with the above training errors

(31)W_i,j(k+1)=W_i,j(k)−η1(∂e_12∂W_i,j+∂e1−2∂W_i,j)W_i,j(k+1)=W_i,j(k)−η2(∂e_22∂W_i,j+∂e2−2∂W_i,j)

where η₁ and η₂ are positive learning rates.

The momentum terms, γ∆W_i,j(k−1) and γΔW¯i,j(k−1) can be added to stabilized the training process. The above Bernstein neural network can be converted to a recurrent (dynamic) form, see Figure 3. The dynamic Bernstein neural network is

(32){ddtx_m(t,α)=P_(t,α)A(x_m(t,α),x_m(t,α))+Q_(t,α)ddtx_m(t,α)=P_(t,α)A(x_m(t,α),x_m(t,α))+Q_(t,α)

Obviously this dynamic network has the form of

f(t,x) =P(t)x+ Q(t)

and it is closed to (2).

The training algorithm is similar as (31), only the training errors are changed as

(33)1){e_1=C1+C2−P_A(x_m,x_m)−Q_e_1=D1+D2−P_A(x_m,x_m)−Q_2){e_2=C1+C2−P_A(x_m,x_m)−Q_e_2=D1+D2−P_A(x_m,x_m)−Q_

4. Applications

In this section, several real applications are used to show how to use the Bernstein neural networks to approximate the solutions of the FDEs. These examples can be expressed by FDEs.

Example 1

The vibration mass system shown in Figure 4 can be modeled by the ODE

(34)ddtx(t)=kmx(t)

where the spring constant is k = 1. The mass m is changeable in [(0.75, 1.125), p(0.7, 0.8,1)], so the position state x(t) has some uncertainties, the ODE (34) can be formed into a FDE based on Z-number. It has the same form as (34), only x(t) becomes a Z-number variable. If the initial position is x(0) = [(0.75 + 0.25α,1.125–0.125 α), p(0.8,0.9,1)], α ϵ [0,1], then the exact solutions of the FDE (34) is [48]

(35)x(t,α)=[((0.75+0.25α)et,(1.125−0.125α)et),p(0.8,0.9,1)]

where t ∈[0,1]. Now the static Bernstein neural network (28) is used, noted as SNN to approximate the Z-number solution [(x_m(t,α),x¯m(t,α)),p(0.8,0.94,1)] (35) where

{x_m(t,α)=(0.75+0.25α)+t∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−jx_m(t,α)=(1.125−0.125α)+t∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−j

Also dynamic Bernstein neural network (32) is used, noted as DNN to approximate the solutions. The learning rates are η = 0.01, γ = 0.01. To compare the results, two other popular methods are used namely Max-Min Euler method and Average Euler method [28]. The comparison results are shown in Table 1 and Table 2. Corresponding solution plots are shown in Figure 5. We can see that both the Bernstein neural networks SNN and DNN (our methods), Max-Min Euler method and Average Euler method can approximate the solutions of the FDEs. The approximation errors of the DNN is much smaller than the other methods.

Table 1.

Solutions of different methods based on Z-numbers.

	SNN	DNN	Max-Min Euler	Average Euler
0	[(0.0684,0.1251),p(0.7,0.8,0.85)]	[(0.0231,0.0671),p(0.7,0.85,0.87)]	[(0.1064,0.1596),p(0.7,0.8,0.85)]	[[(0.2404,0.5138),p(0.6,0.8,0.85)]
0.2	[(0.0735,0.1192),p(0.7,0.8,0.9)]	[(0.0266,0.0675),p(0.75,0.8,0.9)]	[(0.1127,0.1551),p(0.7,0.8,0.87)]	[(0.1588,0.4286),p(0.7,0.8,0.85)]
0.6	[(0.0855,0.1095),p(0.8,0.87,0.95)]	[(0.0339,0.0689),p(0.8,0.9,1)]	[(0.1253,0.1462),p(0.7,0.85,0.9)]	[(0.0082,0.2798),p(0.7,0.81,0.9)]
0.8	[(0.0833,0.0939),p(0.8,0.91,1)]	[(0.0345,0.0526),p(0.8,0.94,1)]	[(0.1247,0.1345),p(0.8,0.9,1)]	[(0.0628,0.2009),p(0.75,0.9,1)]
1	[(0.1029,0.1029),p(0.7,0.8,0.9)]	[(0.0572,0.0572),p(0.8,0.85,0.95)]	[(0.1410,0.1410),p(0.7,0.8,0.87)]	[(0.1410,0.1410),p(0.7,0.8,0.87)]

Table 2.

Approximation errors based on Z-numbers

The following relation is used to transfer the Z-numbers to fuzzy numbers,

α=∫xπP˜(x)dx∫πP˜(x)dx

Consider Z = (A, p) = [(2.1858, 3.2787), p(0.8,0.87,0.95)], then Z^α = [2.1858, 3.2787; 0.87] and so Z′= [0.87 2.1858,0.87 3.2787] . The comparison results of different methods for the fuzzy numbers are shown in Table 3.

α	Exact solution	SNN	DNN	Max-Min Euler	Average Euler
0	[2.0387,3.0581]	[1.9703,3.0043]	[1.9901,3.0305]	[1.9453,2.5980]	[2.2441,2.6191]
0.1	[2.1067,3.0241]	[2.0302,2.9415]	[2.0591,2.9752]	[2.0102,2.8855]	[2.2791,2.6166]
0.2	[2.1746,2.9901]	[2.1059,2.9131]	[2.1283,2.9399]	[2.0750,2.8531]	[2.3140,2.6140]
0.3	[2.2426,2.9561]	[2.1618,2.8707]	[2.1901,2.8931]	[2.1398,2.8207]	[2.3490,2.6115]
0.4	[2.3105,2.9222]	[2.2307,2.8453]	[2.2601,2.8799]	[2.2047,2.7883]	[2.3840,2.6090]
0.5	[2.3785,2.8882]	[2.2984,2.8088]	[2.3288,2.8337]	[2.2695,2.7559]	[2.4189,2.6064]
0.6	[2.4465,2.8542]	[2.3631,2.7784]	[2.3904,2.7955]	[2.3344,2.7234]	[2.4539,2.6039]
0.7	[2.5144,2.8202]	[2.4292,2.7449]	[2.4555,2.7691]	[2.3992,2.6910]	[2.4888,2.6013]
0.8	[2.5824,2.7862]	[2.4895,2.7067]	[2.5101,2.7302]	[2.4641,2.6586]	[2.5238,2.5988]
0.9	[2.6503,2.7523]	[2.5564,2.6769]	[2.5821,2.7001]	[2.5289,2.6262]	[2.5588,2.5963]
1	[2.7183,2.7183]	[2.6199,2.6399]	[2.6414,2.6614]	[2.5937,2.5937]	[2.5937,2.5937]

Table 3.

Solutions of different methods based on fuzzy numbers

The Z-numbers increase degree of reliability of the information. The crucial factor is that incorporated information is not only the most generalized representation of information uncomplicated real world but also incorporated with greater narrative power extracted from human cognition perspective compared with fuzzy number. The comparison between the Z-number Z = [(2.1858, 3.2787), p(0.8, 0.87, 0.95)] and fuzzy number [2.0387, 3.0581] is shown in Figure 6. It can be seen that the Z-number incorporates with various information and the solution of the Z-number is more accurate. The membership function for the restriction in the Z-number is μ_{A_z} = [2.1858,3.2787]. It can be in probability form.

Example 2

The heat treatment system in welding can be modeled as [49]:

(36)ddtx(t)=3Ax2(t)

where transfer area A is uncertainty as A = [(1 + α, 3 – α), p(0.8, 0.87, 0.95)] α ∈ [0,1]. So (ex3) is a FDE based on Z-number. If the initial condition is x(0)=[(0.5α,0.21−α+0.5),p(0.8,0.92,1)] , the static Bernstein neural network (28) has the form of

{x_m(t,α)=0.5α+t∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−jx_m(t,α)=0.21−α+0.5+t∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−j

where the approximate Z-number solution is termed as [(x_m(t,α),x¯m(t,α)),p(0.8,0.9,1)] . With the learning rates η = 0.002 and γ = 0.002 the approximation results for Z-numbers are shown in Table 4. The results of Bernstein neural networks approximation for the fuzzy numbers are shown in Table 5.

α	SNN	DNN
0	[(0.0582,0.0859),p(0.7,0.8,0.85)]	[(0.0250,0.0425),p(0.7,0.82,0.9)]
0.1	[(0.0449,0.0696),p(0.7,0.8,0.9)]	[(0.0224,0.0399),p(0.75,0.82,0.9)]
0.2	[(0.0419,0.0619),p(0.8,0.92,1)]	[(0.0207,0.0394),p(0.8,0.94,1)]
0.3	[(0.0250,0.0348),p(0.7,0.81,0.9)]	[(0.0226,0.0344),p(0.8,0.85,0.96)]
0.4	[(0.0487,0.0689),p(0.7,0.8,0.88)]	[(0.0271,0.0510),p(0.75,0.82,0.9)]
0.5	[(0.0534,0.0665),p(0.8,0.9,1)]	[(0.0160,0.0271),p(0.81,0.92,1)]
0.6	[(0.0494,0.0765),p(0.8,0.9,1)]	[(0.0201,0.0413),p(0.81,0.92,1)]
0.7	[(0.0630,0.0859),p(0.75,0.82,0.9)]	[(0.0303,0.0476),p(0.82,0.9,1)]
0.8	[(0.0393,0.0536),p(0.8,0.92,1)]	[(0.0164,0.0379),p(0.82,0.94,1)]
0.9	[(0.0422,0.0669),p(0.8,0.9,1)]	[(0.0212,0.0430),p(0.8,0.94,1)]
1	[(0.0443,0.0443),p(0.7,0.8,0.88)]	[(0.0186,0.0186),p(0.7,0.82,0.9)]

Table 4.

Bernstein neural networks approximate the Z-numbers

α	SNN	DNN
0	[0.0511,0.0754]	[0.0224,0.0381]
0.1	[0.0402,0.0623]	[0.0203,0.0362]
0.2	[0.0398,0.0588]	[0.0197,0.0374]
0.3	[0.0224,0.0312]	[0.0211,0.0321]
0.4	[0.0433,0.0613]	[0.0246,0.0462]
0.5	[0.0507,0.0631]	[0.0152,0.0258]
0.6	[0.0469,0.0726]	[0.0191,0.0392]
0.7	[0.0571,0.0778]	[0.0288,0.0452]
0.8	[0.0373,0.0509]	[0.0157,0.0362]
0.9	[0.0401,0.0635]	[0.0202,0.0408]
1	[0.0394,0.0394]	[0.0167,0.0167]

Table 5.

Bernstein neural networks approximate the fuzzy number

Example 3

A generalized model of a tank system is displayed in Figure 7. Assume I = t + 1 be inflow disturbances of the tank that will generate vibration in liquid level x, here R = 1 will be the flow obstruction that can be curbed using the valve and A = 1 is considered to be cross section of the mentioned tank. The expression in relation to the liquid level considering the tank can be described as [50]:

(37)ddtx(t)=−1ARx(t)+IA

If the initial condition is x(0) = [(0.96 + 0.04α, 1.01– 0.01 α), p(0.75, 0.82, 0.9)], the static Bernstein neural network (28) has the form of

{x_m(t,α)=(0.96+0.04α)+t∑i=0N∑j=0MλiγjW_i,jti(T−ti)N−iαj(1−αj)M−jx¯m(t,α)=(1.01−0.01α)+t∑i=0N∑j=0MλiγjW¯i,jti(T−ti)N−iαj(1−αj)M−j

where t ∈ [0,1] and the approximate Z-number solution is termed as [(x_m(t,α),x¯m(t,α)),p(0.75,0.81,0.95)] . Also dynamic Bernstein neural network (38) is used to approximate the solutions. To compare the results, the other generalization of neural network method [31] is used. The comparison results for Z-numbers are shown in Table 6. The specifications quoted here are η = 0.001 and γ = 0.001. Corresponding error plots are shown in Figure 8. These errors are the differences of the exact and the approximation solutions, for three different methods: SNN, DNN and NN Z-numbers. The results of Bernstein neural networks approximation for the fuzzy numbers are shown in Table 7. DNN is more accurate than the SNN and the other generalization of neural network method.

α	SNN	DNN	Neural network
0	[(0.0435, 0.0994),p(0.72,0.81,0.87)]	[(0.0112, 0.0442),p(0.75,0.82,0.88)]	[(0.0798, 0.1153),p(0.7,0.75,0.85)]
0.2	[(0.0504, 0.0940),p(0.7,0.8,0.9)]	[(0.0248, 0.0635),p(0.75,0.82,0.9)]	[(0.0878, 0.1375),p(0.7,0.8,0.85)]
0.4	[(0.0441, 0.0802),p(0.8,0.85,0.92)]	[(0.0131, 0.0422),p(0.8,0.9,1)]	[(0.1105, 0.1592),p(0.75,0.83,0.9)]
0.6	[(0.0178, 0.0423),p(0.8,0.92,1)]	[(0.0121, 0.0384),p(0.81,0.94,1)]	[(0.0613, 0.0915),p(0.8,0.9,1)]
0.8	[(0.0608, 0.0709),p(0.71,0.8,0.9)]	[(0.0154, 0.0309),p(0.8,0.87,0.95)]	[(0.0739, 0.0925),p(0.7,0.8,0.85)]
1	[(0.0611, 0.0611),p(0.75,0.82,0.91)]	[(0.0335, 0.0335),p(0.8,0.87,0.92)]	[(0.1007, 0.1007),p(0.7,0.8,0.9)]

Table 6.

Solutions of different methods based on Z-numbers

α	SNN	DNN	Neural network
0	[0.0387, 0.0884]	[0.0101, 0.0398]	[0.0701, 0.1012]
0.2	[0.0451, 0.0841]	[0.0225, 0.0575]	[0.0771, 0.1207]
0.4	[0.0407, 0.0740]	[0.0125, 0.0401]	[0.1001, 0.1442]
0.6	[0.0169, 0.0402]	[0.0115, 0.0365]	[0.0582, 0.0868]
0.8	[0.0544, 0.0635]	[0.0144, 0.0289]	[0.0649, 0.0812]
1	[0.0554, 0.0554]	[0.0311, 0.0311]	[0.0901, 0.0901]

Table 7.

Solutions of different methods based on fuzzy numbers

5. Conclusions

In this paper, two types of Bernstein neural networks are used namely static and dynamic models to approximate the solutions of FDEs on the basis of Z-numbers. Initially the FDE is transformed into four ODEs with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z-number variables, the neural networks are trained. Some real examples are employed to show the effectiveness of the proposed approximation methods with the Bernstein neural networks. The future works are the application of these mentioned methodologies for fuzzy partial differential equations on the basis of Z-numbers.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 10 - 1
Pages: 1226 - 1237
Publication Date: 2017/09/19
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TY  - JOUR
AU  - Raheleh Jafari
AU  - Wen Yu
AU  - Xiaoou Li
AU  - Sina Razvarz
PY  - 2017
DA  - 2017/09/19
TI  - Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks
JO  - International Journal of Computational Intelligence Systems
SP  - 1226
EP  - 1237
VL  - 10
IS  - 1
SN  - 1875-6883
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