universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural...

Fuzzy Sets and Systems 119 (2001) 313–320www.elsevier.com/locate/fss

Universal approximations of continuous fuzzy-valued functionsbymulti-layer regular fuzzy neural networks(

Puyin LiuDepartment of System Engineering and Mathematics, National University of Defence Technology, Changsha, Hunan 410073,

People’s Republic of China

Received 23 July 1998; received in revised form 10 March 1999; accepted 1 July 1999

Abstract

The fact that four-layer feedforward regular fuzzy neural networks with sigmoid function in the �rst hidden layer arecapable of approximately representing continuous fuzzy valued functions on any compact set of R is shown. At �rst, Bernsteinpolynomials associated with fuzzy valued functions are employed to approximate continuous fuzzy valued function de�ned ona compact set. Secondly, by the conclusions related to standard feedforward networks, universal approximations of continuousfuzzy valued functions by regular fuzzy neural networks are obtained. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Regular fuzzy neural networks; Fuzzy-valued polynomials; Universal approximations; Universal approximator

1. Introduction

It is shown that a three-layer feedforward neu-ral network with nonlinear activation function in thehidden layer is capable of approximating generic classof functions, including continuous and integrable ones[13]. These achievements have not only solved theapproximation representation of some multi-variablefunctions by the combination of �nite compositions ofone-variable functions, but are also found to be usefulin many real �elds, such as approximation of struc-tural synthesis [11], system identi�cation [3], patternclassi�cation [4], and adaptive �ltering [7], etc.Recently, some authors have begun to pay atten-

tions to the similar approximation problems in fuzzy

( This work was supported by a grant from National NaturalScience Foundation.

E-mail address: [email protected] (P. Liu).

environments [1,2,9,10]. In [1], Buckley shows thatregular fuzzy neural networks (RFNNs) are notcapable of approximating continuous fuzzy functionsto any degree of accuracy on the compact sets ofF0(R). Based on these conclusions, the authors in[9,10] show some fuzzy functions that can arbitrarilybe closely approximated by RFNNs. Using �-cuts andinterval arithmetic, Buckley and Hayashi [2] demon-strate the fact that neural networks with non-negativeor non-positive fuzzy number inputs are universalapproximators for a class of fuzzy functions ob-tained by the extension principle. Further problems,such as, what are the conditions for fuzzy functionsthat can be approximated to any degree of accu-racy by RFNNs? How can we study approximationcapabilities of RFNNs? etc., are very important andmeaningful mathematical problems in theoretical andapplied areas.

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00132 -3

314 P. Liu / Fuzzy Sets and Systems 119 (2001) 313–320

In this paper, we obtain the universal approximationof continuous fuzzy-valued function F :R→F0(R)by the RFNNs with real input and fuzzy valued output.In addition, we give some equivalent conditions forthe fact that fuzzy-valued functions can abitrarily beclosely approximated by a class of RFNNs. Finally,some important problems related to the subject forfuture research are put forward.

2. Preliminary

Let F(R) be the set of all fuzzy sets on the realnumber set R; N the set of all natural numbers, andF0(R) the set of all fuzzy numbers, i.e. the followingconditions hold for A∈F0(R):(i) ∀�∈ (0; 1]; A�,{x∈R | A(x)¿�} is the closed

interval of R;(ii) Supp( A),{x∈R | A(x)¿0}⊂R is the bound-

edly closed subset of R;(iii) A is normal, i.e. {x∈R | A(x)= 1} 6= ∅:For simplicity, let A0; zero-cut of A, denote thesupport Supp(A) of A. Obviously, A0 is a boundedlyclosed interval of R. Write A�• ,{x∈R | A(x)¿�}.De�ne the metric on F0(R) as follows: for

A; B∈F0(R);

D(A; B)=∨

�∈[0;1](dH(A�; B�))=

∨�∈[0;1]

(dH(A�• ; B�• ));

where dH means Hausdor� metric, i.e. if A; B⊂R;

dH(A; B)

= max

∨x∈A

∧y∈B

|x−y|;∨

y∈B

(∧x∈A

|x−y|)

and∨;∧denote, respectively, the supremum oper-

ator sup(max) and in�mum operator inf(min). It iswell known that (F0(R); D) is a completely separablemetric space [6].Assume A∈F0(R); if A�= [a1�; a2�] for each �∈

[0; 1]; henceforth we write

|A|,∨

�∈[0;1](|a1�| ∨ |a2�|):

By the extension principle [12], each functionf :Rn→Rmay be extended to oneF0(R)n→F(R);

which is still denoted by f for simplicity:

∀A1; : : : ; An ∈F0(R);

f(A1; : : : ; An)(y) =∨

f(x1 ;:::; xn)=y

(n∧i=1

{Ai(xi)}): (1)

For A; B∈F0(R); and c∈R; addition A + B; multi-plication A · B and multiplication by a scalar c; c · Aare de�ned by the extension principle. The followingconclusion follows from [12].

Remark 1. Suppose A; B∈F0(R); c∈R; then theequalities

(A+ B)� = A� + B�; (A · B)� = A� · B�;(c · A)�= c · A�

(2)

hold.

3. Universal approximations of fuzzy-valuedfunctions

Let C denote a subset of the set of all fuzzy-valuedfunctions that R→F0(R): C is said to be continu-ous if given arbitrarily as F ∈C; F is continuous onR. C is called the additive class if H1 + H2 ∈C forarbitrary H1; H2 ∈C; where (H1 + H2)(x)=H1(x) +H2(x):

De�nition 1. Suppose fuzzy-valued function F :R→F0(R): For arbitrary compact set U ⊂R and ”¿0;there is H ∈C; such that ∀x∈U; D(F(x); H (x))¡”:Then C is called the universal approximator for F; Cis also called the universal approximation of F:

The following remark is trivial.

Remark 2. Let C1;C2 be subsets of the set of func-tions that R→F0(R); and ∀H ∈C1;C2 is universalapproximator for H . Then, if C1 is a universal approx-imator for F :R→F0(R); so is C2:

Lemma 1. Assume that I is an arbitrary in-dex set; {ai | i∈ I}; {bi | i∈ I}⊂ [0; 1]; and h¿0;∀i∈ I; |ai − bi |6h: Then the following hold:(i) |∨i∈I ai −∨i∈I bi |6h;(ii) |∧i∈I ai −∧i∈I bi |6h:

P. Liu / Fuzzy Sets and Systems 119 (2001) 313–320 315

Proof. (i) By the hypotheses, if �,∨i∈I ai; �,∨

i∈I bi, then �; �∈ [0; 1]: It su�ces to prove |�−�|6h:∀”¿0; there are i0; j0 ∈ I; such that ai06�¡ai0 + ”;bj06�¡bj0 + ”: Consequently,

ai0 − bj0 − ”¡� − �¡ai0 − bj0 + ”: (3)

If i0 = j0; we obtain by the assumptions and (3) that

− h− ”6ai0 − bj0 − ”¡�− �¡ai0 − bj0 + ”6h+ ”;

i.e. |�− �|¡h+ ”: So |�− �|6h: If i0 6= j0; it suf-�ces to show |�− �|6h in the following cases:(i) ai0¿aj0 ; bi0¿bj0 ; (ii) ai0¿aj0 ; bi0¡bj0 ; (iii) ai0¡aj0 ; bi0¿bj0 ; (iv) ai0¡aj0 ; bi0¡bj0 .In case (i), it is obvious that

− h6aj0 − bj06ai0 − bj0¡ai0 − (�− ”)6ai0 − bi0 + ”6h+ ”:

So (3) implies the fact that − h − ”¡� − �¡h + 2”holds, i.e. |� − �|¡h+ 2”⇒|� − �|6h:In case (ii), we may easily prove that

− h6aj0 − bj06ai0 − bj0¡ai0 − bi06h:

Hence, (3) implies that − h − ”¡� − �¡h + ”⇒|� − �|6h:For the same reasons,we have |� − �|6h in cases

(iii) and (iv). Thus (i) holds.(ii) If ∀i∈ I; a′i =1− ai; b′i =1− bi; we have

a′i ; b′i ∈ [0; 1]; and |a′i − b′i |6h (i∈ I): So by (i),

|∨i∈I a′i −∨i∈I b′i |6h holds. Therefore,∣∣∣∣∣∧i∈Iai −

∧i∈Ibi

∣∣∣∣∣=∣∣∣∣∣(1−

∧i∈Iai

)−(1−

∧i∈Ibi

)∣∣∣∣∣=

∣∣∣∣∣∨i∈Ia′i −

∨i∈Ib′i

∣∣∣∣∣6hi.e. (ii) holds.

By the fact that a∨ b= 12(a+ b+ |a− b|) for

a; b∈R; the following lemma is obvious.

Lemma 2. Suppose n∈N; and a1; : : : ; an; b1; : : : ; bn∈R; then (∑n

i=1 ai) ∨ (∑n

i=1 bi)6∑n

i=1(ai ∨ bi):

Lemma 3. Assume A; A1; A2 ∈F0(R); n∈N; and{Wk | k =1; : : : ; n}; {Vk | k =1; : : : ; n}⊂F0(R):Then;(i) D(A · A1; A · A2)6| A| ·D(A1; A2);(ii) D(

∑nk=1 Wk ;

∑nk=1 Vk)6

∑nk=1D(Wk ; Vk):

Proof. ∀�∈ [0; 1]; set

(Ai)�= [a1i�; a2i�] (i=1; 2); A�= [a1�; a

2�];

(Wk)�= [w1k�; w2k�]; (Vk)�= [v1k�; v

2k�]

(k =1; : : : ; n):

By the assumptions, the following

dH((A1)�; (A2)�) = dH([a11�; a21�]; [a

12�; a

22�])

= |a21� − a22�| ∨ |a11� − a12�|

hold. What one has to do is to show (i), (ii),respectively.(i) Since by Remark 1, ∀�∈[0; 1]; (A · A1)�=

A� · (A1)�; so if (A · A1)�= [�1; �2]; we have

�1 = (a1� · a11�)∧ (a2� · a11�)∧ (a1� · a21�)∧ (a2� · a21�);�2 = (a1� · a11�)∨ (a2� · a11�)∨ (a1� · a21�)∨ (a2� · a21�):

Similarly, if set (A · A2)�= [�1; �2]; the following

�1 = (a1� · a12�)∧ (a2� · a12�)∧ (a1� · a22�)∧ (a2� · a22�);�2 = (a1� · a12�)∨ (a2� · a12�)∨ (a1� · a22�)∨ (a2� · a22�)

hold, and we may easily obtain

|a1� · a11�− a1� · a12�)|6|a11�− a12�| · |A|6|A| ·D(A1; A2);|a2� · a11�− a2� · a12�)|6|a11�− a12�| · |A|6|A| ·D(A1; A2);|a1� · a21�− a1� · a22�)|6|a21�− a22�| · |A|6|A| ·D(A1; A2);|a2� · a21�− a2� · a22�)|6|a21�− a22�| · |A|6|A| ·D(A1; A2):

(4)

So, by Lemma 1 and (4), the following

dH((A · A1)�; (A · A2)�) = dH([�1; �2]; [�1; �2])= |�1 − �1| ∨ |�2 − �2|6 |A| ·D(A1; A2)


hold. Thus, D(A · A1; A · A2)6|A| ·D(A1; A2): Hence,(i) holds.(ii) ∀�∈ [0; 1]; the following hold by Lemma 2:

dH

((n∑k=1

Wk

)�

;

(n∑k=1

Vk

)�

)

= dH

([n∑k=1

w1k�;n∑k=1

w2k�

];

[n∑k=1

v1k�;n∑k=1

v2k�

])

=

∣∣∣∣∣n∑k=1

(w1k� − v1k�)∣∣∣∣∣ ∨

∣∣∣∣∣n∑k=1

(w2k� − v2k�)∣∣∣∣∣

6

(n∑k=1

|w1k� − v1k�|)

∨(

n∑k=1

|w2k� − v2k�|)

6n∑k=1

(|w1k� − v1k�| ∨ |w2k� − v2k�|)

=n∑k=1

dH((Wk)�; (Vk)�):

So

D

(n∑k=1

Wk ;n∑k=1

Vk

)

6∨

�∈[0;1]

(n∑k=1

dH((Wk)�; (Vk)�)

)

6n∑k=1

D(Wk ;Vk);

hence (ii) holds.

By Lemma 3, if f :R→R is continuous, A∈F0(R); and A ·f :R→F0(R) is de�ned as follows:(A ·f)(x)=f(x) · A (x∈R); then A ·f is continuouson R:

Theorem 1. Assume that Fi :R→F0(R) (i=1; 2;: : : ; n) are fuzzy valued functions; and C is the uni-versal approximator respectively for F1; : : : ; Fn: Then;if C is the additive class; C is also the universalapproximator for F1 + · · ·+ Fn:

Proof. By the inductive method, it su�ces to showthe conclusion when n=2: Given an arbitrarily com-pact set U ⊂R and ”¿0; by the assumptions that C isthe universal approximator for F1 and F2; respectively,there are H1; H2 ∈C; such that

∀x∈U; D(F1(x); H1(x))¡”2;

D(F2(x); H2(x))¡”2:

(5)

Set H =H1 + H2; then H ∈C, and by Lemma 3,∀x∈U; the following hold:

D((F1 + F2)(x); H (x))

= D(F1(x) + F2(x); H1(x) + H2(x))

6D(F1(x); H1(x)) + D(F2(x); H2(x))

¡”2+”2= ”:

So, C is the universal approximator for F1 + F2:

By Theorem 1, it is obvious that if C is theuniversal approximator, respectively, for fuzzy func-tions F1; : : : ; Fn; then C is also the universal approx-imator for the linear combination c1 ·F1 + · · · +cn ·Fn (c1; : : : ; cn ∈R):If C is the approximator for F; the properties of F

are similar to those of C:

Theorem 2. Let F :R→F0(R) be a fuzzy valuedfunction; and C be the universal approximator for F;then if C is continuous; F is continuous on R:

Proof. Suppose the conclusion is false, then there arex0 ∈R and ”0¿0; such that ∀�¿0; there is x′ ∈R:|x0− x′|¡�;D(F(x′); F(x0))¿”0: Select, respectively,�=1; 1=2; : : : ; 1=n; : : : ; correspondingly, we obtain thesequence {xn | n∈N}⊂R; such that ∀n∈N; |xn − x0|¡1=n; D(F(xn); F(x0))¿”0: Set U={x0; x1; : : : ; xn;: : :}:Obviously,U ⊂R is a compact set by the fact thatxn→ x0 (n→+∞): The assumptions imply that thereis an H ∈C; such that ∀x∈U; D(F(x); H (x))¡”0=4:On the other hand, H is continuous at x0 by the conti-nuity of C; so there is �0¿0 satisfying the followingfacts:

∀x∈U; |x − x0|¡�0⇒D(H (x); H (x0))¡”04:


Let n0 ∈N satisfying 1=n0¡�0; thusD(H (xn0 ); H (x0))¡”0=4: Therefore,

”06D(F(xn0 ); F(x0))

6D(F(x0); H (x0)) + D(H (x0); H (xn0 ))

+D(H (xn0 ); F(xn0 ))

¡”04+”04+”04¡”0:

This is a contradiction. So, F is continuous on R:

Let

P=

{P |P(x)=

n∑i=1

pi(x) · Ai; n∈N; A1; : : : ; An ∈

F0(R); pi is a polynomial with respect to x}:

Each element P in P is called the fuzzy-valuedpolynomial. Obviously, P is continuous.To show universal approximations of continuous

fuzzy-valued functions by P; we at �rst prove thefollowing lemma.

Lemma 4. Suppose closed interval [a; b]⊂R; andF : [a; b]→F0(R) is continuous. De�ne the realfunction f : [a; b]× [a; b]→R as follows: ∀x; y∈[a; b]; f(x; y)=D(F(x); F(y)): Then f is continuouson [a; b]× [a; b]:

Proof. Given arbitrarily (x0; y0)∈ [a; b] × [a; b]; and”¿0: By the continuities of F at x0 and y0; respec-tively, there is a �¿0; such that

∀x; y∈ [a; b]; |x − x0|¡�⇒D(F(x); F(x0))¡”2;

|y − y0|¡�⇒D(F(y); F(y0))¡”2:

So, ∀(x; y)∈ [a; b]× [a; b]; if |x− x0|¡�; |y−y0|¡�;we have by the triangle inequality of metric:

D(F(x); F(y))

6D(F(x0); F(y)) + D(F(x0); F(x))

6D(F(x0); F(y0)) + D(F(y0); F(y))

+D(F(x0); F(x));

which implies D(F(x); F(y))−D(F(x0); F(y0))6D(F(x0); F(x))+D(F(y0); F(y)): For the samereason, the following holds:

D(F(x0); F(y0))

6D(F(x0); F(y)) + D(F(y); F(y0))

6D(F(x); F(x0)) + D(F(x); F(y))

+D(F(y); F(y0));

which implies D(F(x0); F(y0))−D(F(x); F(y))6D(F(x0); F(x)) + D(F(y0); F(y)): Therefore,

|D(F(x); F(y))− D(F(x0); F(y0))|6D(F(y0); F(y)) + D(F(x0); F(x));

i.e.

|D(F(x); F(y))− D(F(x0); F(y0))|¡”2+”2= ”:

So, f is continuous at (x0; y0); which proves thelemma.

Theorem 3. Suppose F :R→F0(R) is continuousfuzzy valued function; then for an arbitrary bound-edly closed interval [a; b] and ”¿0; there is a P ∈P;such that ∀x∈ [a; b]; D(F(x); P(x))¡”:

Proof. It will not harm to assume [a; b] = [0; 1]:Since F is continuous on [a; b]; F is uniformlycontinuous on [a; b]: Consequently, there is a �¿0;such that ∀x; y∈ [0; 1]; if |x−y|¡�; we haveD(F(x); F(y))¡”=2: So if there is an n∈N; thefollowing

∀k =1; : : : ; n; ∀x; y∈[k − 1n;kn

];

D(F(x); F(y))¡”2

holds. Let

P(x)=n∑i=0

(ni

)xi(1− x)n−i ·F

(in

)(x∈ [0; 1]):

Then P ∈P: By Lemma 4, f(x; y),D(F(x); F(y)) iscontinuous on [0; 1]× [0; 1]; hence there is an M¿0;such that ∀x; y ∈ [0; 1]; D(F(x); F(y))6M:We may


easily show that

n∑i=0

(ni

)(i − nx)2xi(1− x)n−i

= nx(1− x)6n2

(x ∈ [0; 1]):

So, considering the fact that

n∑i=0

(ni

)xi(1− x)n−i=1;

n∑|x−i=n|¡�

(ni

)xi(1− x)n−i61

and Lemma 3, we obtain for x∈ [0; 1] that

D(F(x); P(x))

6n∑i=0

D((n

i

)xi(1− x)n−i ·F(x);

(ni

)xi(1− x)n−i ·F

(in

))

6∑

|x−i=n|¡�

(ni

)xi(1− x)n−iD

(F(x); F

(in

))

+∑

|x−i=n|¿�

(ni

)xi(1− x)n−iD

(F(x); F

(in

))

6”2+

∑|x−i=n|¿�

(x − i=n�

)2(ni

)xi(1− x)n−i

×D(F(x); F

(in

))6”2+

M2n�2

: (6)

Let n¿M=�2”, then ∀x∈ [0; 1]; D(F(x); P(x))¡”:

Obviously, Theorem 3 holds if we substitute a com-pact set U ⊂R for [a; b]:

4. Approximation capability of a class of RFNNs

In this section, we introduce four-layer feedforwardRFNNs, by which universal approximations of con-tinuous fuzzy valued functions may be investigated.De�ne

Fig. 1. Two hidden layer feedforward RFNN.

H=

H |H (x)=

n∑i=1

Wi

m∑j=1

Vij · �(x · Uj + �j)

n; m∈N; Wi; Vij ; Uj; �j ∈F0(R)

;

where � is the extended function of the given function� :R→R:

Remark 3. ∀H ∈H; H is a four-layer feedforwardRFNN with activation function �; threshlod vector(�1; : : : ; �m) in the �rst hidden layer. The topologicalarchitecture is as shown in Fig. 1.

Restricting fuzzy numbers Uj; Vij ; �j, respectively,to be uj; vij; �j ∈R; we obtain the subsetH0 ofH:

H0 =

H |H (x)=

n∑i=1

Wi

m∑j=1

vij · �(uj · x + �j)

n; m∈N; Wi ∈F0(R); vij; uj; �j ∈R :

Obviously, both H and H0 are additive classes,and if � :R→R is continuous on R; H is continuousand so isH0:

Theorem 4. Let one of the following conditions holdfor � :R→R:(i) � is a sigmoid function; i.e. � is bounded and

limx→−∞ �(x)= 0; limx→+∞ �(x)= 1;


(ii) � is a bounded; continuous and nonconstantfunction on R:

Then; given arbitrarily that P ∈P; H0 is the uni-versal approximator for P; consequently; so isH:

Proof. Given arbitrarily that ”¿0; and compact setU ⊂R; let P(x)= ∑n

i=1 pi(x) · Ai (x∈R): Since fori=1; : : : ; n; pi is a polynomial, so pi is continuous onR: Therefore, by the conclusions in [5,8], there areK ∈N and vj; uj; �j ∈R (j=1; : : : ; K); such that eithercondition (i) or (ii) holds, the following hold:

∀x∈U;∣∣∣∣∣∣pi(x)−

K∑j=1

vj · �(uj · x + �j)∣∣∣∣∣∣¡

”

|Ai|:

(7)

Set Hi(x)=(∑K

j=1 vj · �(uj · x + �j))· Ai (x∈R):

Then Hi ∈H0; and the following hold for each x∈Uby Lemma 3 and (7):

D(pi(x) · Ai; Hi(x))

= D

pi(x) · Ai;

K∑j=1

vj · �(uj · x + �j) · Ai

6|Ai| ·∣∣∣∣∣∣pi(x)−

K∑j=1

vj · �(uj · x + �j)∣∣∣∣∣∣¡”:

So, H0 is the universal approximator for Fi: Fi(x)=pi(x) · Ai (i=1; : : : ; n): By the fact that H0 is anadditive class and Theorem 1, H0 is also the uni-versal approximator for P: P(x)=

∑ni=1 Fi(x)=∑n

i=1 pi(x) · Ai; and so isH by the fact thatH0⊂H:

Theorem 5. Let � :R→R be bounded; continuousand nonconstant; and F :R→F0(R) be continuous.ThenH0 is the universal approximator for F; so isH:

Proof. By Theorem 3, P is the universal approxima-tor for F: Theorem 4 implies that for each P ∈P; H0

is the universal approximator to P: Consequently, thetheorem holds by Remark 2.

Theorem 6. Suppose � :R→R is a continuous;bounded and nonconstant function; and F :R→

F0(R) is a fuzzy valued function; then the followingare equivalent:(i) F is continuous;(ii) H0 is the universal approximator for F ;(iii) H is the universal approximator for F ;(iv) P is the universal approximator for F:

Proof. Theorem 5 implies that (i)⇒ (ii); (ii)⇒ (iii)is trivial; Theorem 2, and the continuity of H implythat (iii)⇒ (i). By the continuity ofP and Theorems 2and 3, we may prove that (i)⇔ (iv).

Considering the fuzzy-valued functions F1; F2 :R→F0(R):

F1(x)= A1 sin(x · B1) + · · ·+ An sin(x ·Bn);F2(x)= A1 cos(x ·B1) + · · ·+ An cos(x ·Bn)for A1; : : : ; An; B1; : : : ; Bn ∈F0(R) are continuous onR, we obtain by Theorem 6 thatH0; as well asH isthe universal approximator for F1; F2, respectively.

5. Conclusions

This paper discusses the approximation capabilitiesof multi-layer feedforward RFNNs to fuzzy valuedfunctions. If the activation function in the �rst hiddenlayer is continuous, bounded and non-constant, someequivalent conditions for fuzzy valued functions thatcan be arbitrarily closely approximated by the RFNNsare shown. The conclusions obtained here general-ize the ones in [1,2,10]. For the same reasons, wemay obtain similar conclusions for the multi-variablefuzzy valued functions that Rd→F0(R): The fur-ther problems for future research are the universal ap-proximations of fuzzy functions thatF0(R)d→F0(R)by RFNNs. Can some equivalent conditions be con-structed for such functions?

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