Simulation and evaluation of system of fuzzy linear Fredholm integro-differential equations with fuzzy neural network

Abstract

In this paper, fuzzy neural network (FNN) can be trained with crisp and fuzzy data. The work of this paper is an expansion of the research of fuzzy linear Fredholm integro-differential equations. In this work, we interpret a fuzzy integro-differential equations. In this paper, a novel hybrid method based on FNN and Newton–Cotes methods with positive coefficient for the solution of system of fuzzy linear Fredholm integro-differential equations of the second kind with fuzzy initial values is presented. The problem formulation of the proposed UA is quite straightforward. To obtain the “best-approximated” solution of system of fuzzy linear Fredholm integro-differential equations, the adjustable parameters of PFNN are systematically adjusted by using the learning algorithm. Within this paper, the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Based on the extension principle, a simple algorithm from the cost function of the fuzzy neural network is proposed, in order to find the approximate parameters. We propose a learning algorithm from the cost function for adjusting of fuzzy weights. Here neural network is considered as a part of a larger field called neural computing or soft computing. Finally, we illustrate our approach by some numerical examples.

Appendices

Appendix 1

Definition 6.1

([21, 22]). Let $f:[a,b]\longrightarrow E^1$, for each partition $p=\{t_0,t_1,\ldots , t_n\}$ of [a, b] and for arbitrary $\xi _i\in [t_{i-1},t_i],1\le i\le n$ suppose

$$\begin{aligned} \begin{array}{cc} R_p=\sum _{i=1}^nf(\xi _i)(t_i-t_{i-1}),\\ \\ \Delta {:}{=}\mathrm{max}\{|t_i-t_{i-1}|,i=1,2,\ldots ,n\}. \end{array} \end{aligned}$$

The definite integral of f(t) over [a, b] is

$$\begin{aligned} \int _{a}^{b}f(t){\mathrm{d}}t={\mathrm{lim}}_{\Delta \longrightarrow 0}R_p \end{aligned}$$

provided that this limit exists in the metric D.

If the fuzzy function f(t) is continuous in the metric D, its definite integral exists [22] and also

$$\begin{aligned} \begin{array}{cc} \left( \underline{\int _{a}^{b}f(t;r){\mathrm{d}}t}\right) =\int _{a}^{b}{\underline{f}}(t;r){\mathrm{d}}t,\\ \\ \left( \overline{\int _{a}^{b}f(t;r){\mathrm{d}}t}\right) =\int _{a}^{b}{\overline{f}}(t;r){\mathrm{d}}t. \end{array} \end{aligned}$$

Definition 6.2

Let $u,v\in E^1$. If there exists $w\in E^1$ such that $u=v+w$ then w is called the H-difference of u, v and it is denoted by $u-v.$

Definition 6.3

A function $f:(a,b)\longrightarrow E^1$ is called H-differentiable at ${\hat{t}}\in (a,b)$ if, for $h>0$ sufficiently small, there exist the H-differences $f({\hat{t}}+h)-f({\hat{t}}),f({\hat{t}})-f({\hat{t}}-h),$ and an element $f^{\prime }({\hat{t}})\in E^1$ such that:

$$\begin{aligned} {\mathrm{lim}}_{h\longrightarrow 0^+}D\left( \frac{f({\hat{t}}+h)-f({\hat{t}})}{h},f^{\prime }({\hat{t}})\right) = {\mathrm{lim}}_{h\longrightarrow 0^+}D\left( \frac{f({\hat{t}})-f({\hat{t}}-h)}{h},f^{\prime }({\hat{t}})\right) =0. \end{aligned}$$

Then, $f^{\prime }({\hat{t}})$ is called the fuzzy derivative of f at ${\hat{t}}.$

Appendix 2: Derivation of a learning algorithm in PFNN

Let us denote the fuzzy connection weights $V_{pj},\ p=1,\ldots ,n,\ j=1,\ldots ,m$ to the output unit by its parameter values as $V_{pj}=(v_{pj}^{(1)},\ldots ,v_{pj}^{(q)},\ldots ,v_{pj}^{(r)})$. The amount of modification of each parameter value for problem (16) is written as [28]

$$\begin{aligned} v_{ij}^{(q)}(t+1)= & \,v_{ij}^{(q)}(t)+\bigtriangleup v_{pj}^{(q)}(t),\\ \bigtriangleup v_{pj}^{q}(t)= & -\eta \sum _{i=1}^{n}\sum _{k=1}^{g}\frac{\partial e_{ikh}}{\partial v_{pj}^{(q)}}+\alpha .\bigtriangleup v_{pj}^{(q)}(t-1), \end{aligned}$$

where t indexes the number of adjustments, $\eta$ is a learning rate (positive real number) and $\alpha$ is a momentum term constant (positive real number).

Thus, our problem is to calculate the derivative $\frac{\partial e_{ikh}}{\partial v_{pj}^{(q)}}$. Let us rewrite $\frac{\partial e_{ikh}}{\partial v_{pj}^{(q)}}$ as follows:

$$\begin{aligned} \frac{\partial e_{ikh}}{\partial v_{pj}^{(q)}}=\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^{L}}.\frac{\partial [V_{ij}]_h^{L}}{\partial v_{pj}^{(q)}}+\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^{U}}.\frac{\partial [V_{pj}]_h^{U}}{\partial v_{pj}^{(q)}}. \end{aligned}$$

In this formulation, $\frac{\partial [V_{ij}]_h^{L}}{\partial v_{ij}^{(q)}}$ and $\frac{\partial [V_{ij}]_h^{U}}{\partial v_{ij}^{(q)}}$ are easily calculated from the membership function of the fuzzy connection weight $V_{ij}$.

On the other hand, the derivatives $\frac{\partial e_{ikh}}{\partial [V_{ij}]_h^L}$ and $\frac{\partial e_{ikh}}{\partial [V_{ij}]_h^U}$ are independent of the shape of the fuzzy connection weight. They can be calculated from the cost function $e_{ikh}$ using the input–output relation of our fuzzy neural network for the h-level sets. When we use the cost function with the weighting scheme in (29), $\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^L}$ and $\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^U},$ are calculated as follows:

[Calculation of $\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^L}$]

If $i=p$ then we have

$$\begin{aligned} \frac{\partial e_{ikh}}{\partial [V_{ij}]_h^{L}}= & \,\delta ^L.\left[ \frac{\partial [N_i(x_k,P_i)]_h^L }{\partial [V_{ij}]_h^{L}}+(x_k-a).\frac{\partial z_{ij}}{\partial x}\right. \\&\left. -\frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^L}{\partial [y_{T_i}(x_k,P_i)]_h^{L}}.\frac{\partial [y_{T_i}(x_k,P_i))]_h^L}{\partial [V_{ij}]_h^{L}}\right] , \end{aligned}$$

where

$$\begin{aligned} \delta ^L= & \left( \left[ \frac{{\mathrm{d}}y_{T_i}(x_k,P_i)}{{\mathrm{d}}x}\right] _h^L-\left[f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))\right]_h^L\right) ,\\ \left. \frac{\partial [y_{T_i}(x_k,P_i))]_h^L}{\partial [V_{ij}]_h^{L}}\right]= & (x_k-a).z_{ij}, \quad \frac{\partial N_i(x_k,P_i)]_h^L }{\partial [V_{ij}]_h^{L}}=z_{ij}, \end{aligned}$$

otherwise

$$\begin{aligned}&\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^{L}}= \delta ^L.\left[ -\frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^L}{\partial [y_{T_p}(x_k,P_i)]_h^{L}}.\frac{\partial [y_{T_p}(x_k,P_i))]_h^L}{\partial [V_{pj}]_h^{L}}\right] . \end{aligned}$$

[Calculation of $\frac{\partial e_{ikh}}{\partial [V_{pj}]_h^U}$]

If $i=p$ then we have

$$\begin{aligned} \frac{\partial e_{ikh}}{\partial [V_{ij}]_h^{U}}= & \,\delta ^U.\left[ \frac{\partial N_i(x_k,P_i)]_h^U }{\partial [V_{ij}]_h^{U}}+(x_k-a).\frac{\partial z_{ij}}{\partial x}\right. \\&\left. - \frac{\partial [f_i(x_k,y_{1T}(x_k,P_1),\ldots ,,y_{nT}(x_k,P_n))]_h^U}{\partial [y_{iT}(x_k,P_i)]_h^{U}}.\frac{\partial [y_{iT}(x_k,P_i))]_h^U}{\partial [V_{ij}]_h^{U}}\right] , \end{aligned}$$

where

$$\begin{aligned} \delta ^U= & \left( \left[ \frac{{\mathrm{d}}y_{T_i}(x_k,P_i)}{{\mathrm{d}}x}\right] _h^U-[f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^U\right) ,\\ \left. \frac{\partial [y_{T_i}(x_k,P_i))]_h^U}{\partial [V_{ij}]_h^{U}}\right]= & (x_k-a).z_{ij}, \quad \frac{\partial N_i(x_k,P_i)]_h^U }{\partial [V_{ij}]_h^{U}}=z_{ij}, \end{aligned}$$

otherwise

$$\begin{aligned} \frac{\partial e_{ikh}}{\partial [V_{pj}]_h^{U}}= \delta ^U.\left[ -\frac{\partial [f_i(x_k,y_{1T}(x_k,P_1),\ldots ,,y_{nT}(x_k,P_n))]_h^U}{\partial [y_{T_p}(x_k,P_i)]_h^{U}}.\frac{\partial [y_{T_p}(x_k,P_i))]_h^U}{\partial [V_{pj}]_h^{U}}\right] , \end{aligned}$$

In our PFNN, the connection weights and biases to the hidden units are real numbers. The nonfuzzy connection weight $w_{pj}$ to the jth hidden unit is updated in the same manner as the parameter values of the fuzzy connection weight $V_{pj}$ as follows:

$$\begin{aligned} w_{pj}(t+1)= & w_{pj}(t)+\bigtriangleup w_{pj}(t),\\ \bigtriangleup w_{pj}(t)= & -\eta \sum _{i=1}^{n}\sum _{k=1}^{g} \frac{\partial e_{ikh}}{\partial w_{pj}} +\alpha \bigtriangleup w_{pj} (t-1). \end{aligned}$$

The derivative $\frac{\partial e_{ikh}}{\partial w_{pj}}$ can be calculated from the cost function $e_{ikh}$ using the input–output relation of our PFNN for the h-level sets. When we use the cost function with the weighting scheme, $\frac{\partial e_{ikh}}{\partial w_{pj}}$ for $i=p$ is calculated as follows:

$$\begin{aligned} \frac{\partial e_{ikh}}{\partial w_{ij}}= & \delta ^L.\left[ \frac{\partial [N_i(x_k,P_i)]_h^L }{\partial w_{ij}}+(x_k-a).[V_{ij}]_h^L.z_{ij}+(x_k-a).x_k.[V_{ij}]_h^L.z_{ij}(1-z_{ij})w_{ij}\right. \\&-(x_k-a).[V_{ij}]_h^L.z_{ij}^2-2(x_k-a).x_k.[V_{ij}]_h^L.z_{ij}^2(1-z_{ij})w_{ij}\\&-\left( \frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^L}{\partial [y_{T_i}(x_k,P_i))]_h^L}\right. \\&\left. \left. .\frac{\partial [y_{T_i}(x_k,P_i)]_h^L}{\partial w_{ij}}+\frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^L}{\partial [y_{T_i}(x,P_i))]_h^U}.\frac{\partial [y_{T_i}(x_k,P_i)]_h^U}{\partial w_{ij}}\right) \right] \\&+\delta ^U.\left[ \frac{\partial N_i(x_k,P_i)]_h^U }{\partial w_{ij}}+(x_k-a).[V_{ij}]_h^U.z_{ij}+(x_k-a).x_k.[V_{ij}]_h^U.z_{ij}(1-z_{ij})w_{ij}\right. \\&-(x_k-a).[V_{ij}]_h^U.z_{ij}^2-2(x_k-a).x_k.[V_{ij}]_h^U.z_{ij}^2(1-z_{ij})w_{ij}\\&-\left( \frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^U}{\partial [y_{T_i}(x_k,P_i))]_h^L}\right. \\&\left. \left. .\frac{\partial [y_{T_i}(x_k,P_i)]_h^L}{\partial w_{ij}}+\frac{\partial [f_i(x_k,y_{T_1}(x_k,P_1),\ldots ,,y_{T_n}(x_k,P_n))]_h^U}{\partial [y_{T_i}(x_k,P_i))]_h^U}.\frac{\partial [y_{T_i}(x_k,P_i)]_h^U}{\partial w_{ij}}\right) \right] , \end{aligned}$$

where

$$\begin{aligned} \frac{\partial [N_i(x_k,P_i)]_h^L }{\partial w_{ij}}= & \frac{\partial [N_i(x_k,P_i)]_h^L }{\partial z_{ij}}.\frac{\partial z_{ij} }{\partial net_{ij}}.\frac{\partial net_{ij}}{\partial w_{ij}}=[V_{ij}]_h^L.z_{ij}.(1-z_{ij}).x_k,\\ \frac{\partial [N_i(x_k,P_i)]_h^U }{\partial w_{ij}}= & \frac{\partial [N_i(x_k,P_i)]_h^U }{\partial z_{ij}}.\frac{\partial z_{ij}}{\partial net_{ij}}.\frac{\partial net_{ij}}{\partial w_{ij}}=[V_{ij}]_h^U.z_{ij}.(1-z_{ij}).x_k,\\ \frac{\partial [y_{T_i}(x_k,P_i))]_h^L }{\partial w_{ij}}= & (x_k-a).\frac{\partial [N_i(x_k,P_i)]_h^L }{\partial w_{ij}},\\ \frac{\partial [y_{T_i}(x_k,P_i))]_h^U }{\partial w_{ij}}= & (x_k-a).\frac{\partial [N_i(x_k,P_i)]_h^U }{\partial w_{ij}}. \end{aligned}$$

The nonfuzzy $w_{pj}$ for $i\ne p$ and nonfuzzy biases to the hidden units are updated in the same manner as above.