# Machine Learning for Modeling the Singular Multi-Pantograph Equations

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## Abstract

**:**

## 1. Introduction

- A new numerical method is proposed for solving singular MDEs.
- For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution.
- A new approach on the basis of the Lyapunov theorem is introduced for convergence and stability analysis.
- Square root cubature Kalman filter is developed for the optimization of the suggested solver.
- Several statistical examinations are presented to demonstrate the accuracy and stability.

## 2. Problem Formulation

## 3. T2-FLS Structure

- (1)
- Get the input t.
- (2)
- The input t is mapped into time range $\left(\right)$.
- (3)
- The $\left(\right)$ is divided into M section and for each section a Gaussian membership function (MF) with mean ${m}_{l},\phantom{\rule{0.166667em}{0ex}}l=1,\dots ,M$ and variance ${\nu}_{l}$ is considered.
- (4)
- The upper and lower firing rules are computed as:$${\overline{f}}_{l}\left(t\right)=\mathrm{exp}\left(\right)open="("\; close=")">-\frac{{\left(\right)open="("\; close=")">t-{m}_{l}}^{}}{2}{\overline{\nu}}_{l}^{2},l=1,\dots ,M,$$$${\underline{f}}_{l}\left(t\right)=\mathrm{exp}\left(\right)open="("\; close=")">-\frac{{\left(\right)open="("\; close=")">t-{m}_{l}}^{}}{2}{\underline{\nu}}_{l}^{2},l=1,\dots ,M.$$
- (5)
- The normalized rule firings (type-reduction by the Nie-Tan approach [39]) are obtained as:$${\psi}_{l}\left(t\right)=\frac{{\overline{f}}_{l}\left(t\right)+{\underline{f}}_{l}\left(t\right)}{{\displaystyle \sum _{l=1}^{M}}{\overline{f}}_{l}\left(t\right)+{\underline{f}}_{l}\left(t\right)}.$$
- (6)
- The output is obtained as:$$\widehat{\chi}\left(\right)open="("\; close=")">t,\theta ,m,\nu ,$$$$\dot{\widehat{\chi}}\left(\right)open="("\; close=")">t,\theta ,m,\nu .$$From (8), ${\dot{f}}_{l}\left(t\right)$ is:$${\dot{f}}_{l}\left(t\right)=-\frac{2\left(\right)open="("\; close=")">t-{m}_{l}}{}{\nu}_{l}^{2}$$Equation (9), can be rewritten as:$${\dot{f}}_{l}\left(t\right)=-\frac{2\left(\right)open="("\; close=")">t-{m}_{l}}{}{\nu}_{l}^{2}$$Then from (8) and (10), $\dot{\widehat{\chi}}\left(\right)open="("\; close=")">t,\theta ,m,\nu $ is rewritten as:$$\dot{\widehat{\chi}}\left(\right)open="("\; close=")">t,\theta ,m,\nu $$Similarly, from (11), $\ddot{\widehat{\chi}}\left(\right)open="("\; close=")">t,\theta ,m,\nu $ is obtained as:$$\begin{array}{c}\ddot{\widehat{\chi}}\left(\right)open="("\; close=")">t,\theta ,m,\nu =\hfill \end{array}$$

## 4. Learning Method

- (1)
- Consider error covariance as ${\varphi}_{k-1}$ at sample time $k-1$ and compute cubature points ${C}_{h}$, $h=1,\dots ,6M$ as:$${C}_{h,k-1}={\varphi}_{k-1}{\varsigma}_{h}+{\xi}_{k-1},$$$${\varsigma}_{h}=\left(\right)open="\{"\; close>\begin{array}{c}\begin{array}{c}\sqrt{3M}{\left[\begin{array}{ccccc}0& \cdots & \stackrel{h-th\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}element}{1}& \cdots & 0\end{array}\right]}^{T}\hfill \\ h=1,2,\dots ,3M\hfill \end{array}\\ \begin{array}{c}\sqrt{3M}{\left[\begin{array}{ccccc}0& \cdots & \stackrel{h-th\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}element}{-1}& \cdots & 0\end{array}\right]}^{T}\hfill \\ h=3M+1,\dots ,6M.\hfill \end{array}\end{array}$$
- (2)
- For each ${w}_{\iota}$ in (16), evaluate the cost function J as:$$\begin{array}{c}{J}_{h}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}{\left(\right)open="("\; close=")">{\ddot{\widehat{\chi}}}_{i}\left(\right)open="("\; close=")">t|{C}_{h}+{\displaystyle \sum _{k=1}^{n}}{\dot{\widehat{\chi}}}_{i}\left(\right)open="("\; close=")">{\gamma}_{k}t|{C}_{h}/{P}_{k,i}\left(t\right)+{\widehat{\chi}}_{i}\left(\right)open="("\; close=")">t|{C}_{h}}^{/}-F\left(t\right)\hfill \\ 2\end{array}$$
- (3)
- From (18), estimate ${J}_{m}$ as the mean of ${J}_{h},\phantom{\rule{0.166667em}{0ex}}h=1,\dots ,6M$:$${J}_{m}=\sum _{h=1}^{6M}{J}_{h}/6M.$$
- (4)
- Define ${Z}_{k-1}$ as:$$\begin{array}{c}{Z}_{k-1}=\frac{1}{\sqrt{6M}}\left(\right)open="["\; close>{J}_{1,k-1}-{J}_{m,k-1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{J}_{2,k-1}-{J}_{m,k-1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$
- (5)
- From (20), compute the square-root of covariance matrix as:$${\varphi}_{zz,k-1}=\mathrm{Tria}\left(\left[\begin{array}{cc}{Z}_{k-1}& {S}_{R,k}\end{array}\right]\right),$$
- (6)
- Compute cross-covariance ${\pi}_{\xi z,k-1}$ as:$${\pi}_{\xi z,k-1}={\zeta}_{k-1}{Z}_{k-1}^{T},$$$$\begin{array}{c}{\zeta}_{k-1}=\frac{1}{\sqrt{6M}}\left(\right)open="["\; close>{C}_{1,k-1}-{\xi}_{k-1},{C}_{2,k-1}-{\xi}_{k-1},\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$
- (7)
- Obtain Kalman gain as:$${\kappa}_{k}=\left(\right)open="("\; close=")">{\pi}_{\xi z,k-1}/{\varphi}_{zz,k-1}^{T}$$
- (8)
- Update $\xi $ as:$${\xi}_{k}={\xi}_{k-1}-{\kappa}_{k}{J}_{m}.$$
- (9)
- Update error covariance as:$${S}_{R,k}=\mathrm{Tria}\left(\left[\begin{array}{cc}{\zeta}_{k-1}-{\kappa}_{k}{Z}_{k-1}& {\kappa}_{k}{S}_{R,k-1}\end{array}\right]\right).$$

## 5. Stability and Convergence Analysis

## 6. Evaluation Index

## 7. Simulations

**Example**

**1.**

**Example**

**2.**

**Remark**

**1.**

**Example**

**3.**

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

SCKF | Square root cubature Kalman filter |

SMDE | Singular multi-pantograph differential equations |

T2-FLS | Type-2 fuzzy logic system |

FLS | Fuzzy logic system |

NN | Neural network |

MDE | Multi-pantograph differential equation |

RMSE | Root mean square error |

TIC | Inequality coefficient of Theil index |

VAR | variance |

FIT | Fitness |

IR | Interquartile range |

Med | Median |

Min | Minimum |

M | Number of rules |

${m}_{l}$ | Center of l-th Gaussian membership function |

${\overline{\nu}}_{l}$ | Upper standard division |

${\underline{\nu}}_{l}$ | Lower standard division |

J | Cost function |

${\psi}_{l}$ | l-th rule firing |

$\varphi $ | Covariance matrix |

$\kappa $ | Kalman gain |

N | Number of data samples |

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**Figure 5.**Example 1: (

**a**): The value of TIC at each iteration; (

**b**): Histogram plot for TIC; (

**c**): Box plot for TIC.

**Figure 6.**Example 1: (

**a**): The value of RMSE at each iteration; (

**b**): Histogram plot for RMSE; (

**c**): Box plot for RMSE.

**Figure 7.**Example 1: (

**a**): The value of VAR at each iteration; (

**b**): Histogram plot for VAR; (

**c**): Box plot for VAR.

**Figure 8.**Example 1: (

**a**): The value of FIT at each iteration; (

**b**): Histogram plot for FIT; (

**c**): Box plot for FIT.

**Figure 11.**Example 2: (

**a**): The value of TIC at each iteration; (

**b**): Histogram plot for TIC; (

**c**): Box plot for TIC.

**Figure 12.**Example 2: (

**a**): The value of RMSE at each iteration; (

**b**): Histogram plot for RMSE; (

**c**): Box plot for RMSE.

**Figure 13.**Example 2: (

**a**): The value of VAR at each iteration; (

**b**): Histogram plot for VAR; (

**c**): Box plot for VAR.

**Figure 14.**Example 2: (

**a**): The value of FIT at each iteration; (

**b**): Histogram plot for FIT; (

**c**): Box plot for FIT.

t | Min | Mean | Med | IR |
---|---|---|---|---|

0 | 0.0218 | 0.0776 | 0.0823 | 0.0451 |

0.0500 | 0.0228 | 0.0786 | 0.0830 | 0.0448 |

0.1000 | 0.0223 | 0.0788 | 0.0835 | 0.0455 |

0.1500 | 0.0200 | 0.0774 | 0.0827 | 0.0473 |

0.2000 | 0.0154 | 0.0736 | 0.0799 | 0.0498 |

0.2500 | 0.0089 | 0.0672 | 0.0746 | 0.0519 |

0.3000 | 0.0014 | 0.0589 | 0.0675 | 0.0523 |

0.3500 | 0.0040 | 0.0505 | 0.0595 | 0.0520 |

0.4000 | 0.0001 | 0.0430 | 0.0514 | 0.0500 |

0.4500 | 0.0030 | 0.0373 | 0.0434 | 0.0416 |

0.5000 | 0.0039 | 0.0323 | 0.0354 | 0.0328 |

0.5500 | 0.0003 | 0.0279 | 0.0278 | 0.0202 |

0.6000 | 0.0006 | 0.0242 | 0.0231 | 0.0139 |

0.6500 | 0.0042 | 0.0211 | 0.0209 | 0.0129 |

0.7000 | 0.0026 | 0.0179 | 0.0167 | 0.0159 |

0.7500 | 0.0001 | 0.0149 | 0.0124 | 0.0138 |

0.8000 | 0.0019 | 0.0121 | 0.0120 | 0.0127 |

0.8500 | 0.0012 | 0.0092 | 0.0091 | 0.0112 |

0.9000 | 0.0000 | 0.0062 | 0.0057 | 0.0087 |

0.9500 | 0.0003 | 0.0033 | 0.0030 | 0.0053 |

1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

t | Min | Mean | Med | IR |
---|---|---|---|---|

0 | 0.0004 | 0.0006 | 0.0006 | 0.0001 |

0.0500 | 0.0006 | 0.0009 | 0.0009 | 0.0002 |

0.1000 | 0.0010 | 0.0014 | 0.0015 | 0.0003 |

0.1500 | 0.0013 | 0.0016 | 0.0016 | 0.0002 |

0.2000 | 0.0000 | 0.0008 | 0.0008 | 0.0005 |

0.2500 | 0.0002 | 0.0014 | 0.0013 | 0.0014 |

0.3000 | 0.0013 | 0.0048 | 0.0047 | 0.0025 |

0.3500 | 0.0038 | 0.0090 | 0.0092 | 0.0038 |

0.4000 | 0.0065 | 0.0133 | 0.0137 | 0.0050 |

0.4500 | 0.0090 | 0.0171 | 0.0176 | 0.0059 |

0.5000 | 0.0112 | 0.0204 | 0.0210 | 0.0067 |

0.5500 | 0.0134 | 0.0234 | 0.0240 | 0.0073 |

0.6000 | 0.0155 | 0.0261 | 0.0268 | 0.0077 |

0.6500 | 0.0175 | 0.0285 | 0.0292 | 0.0079 |

0.7000 | 0.0191 | 0.0301 | 0.0309 | 0.0079 |

0.7500 | 0.0201 | 0.0307 | 0.0316 | 0.0075 |

0.8000 | 0.0206 | 0.0306 | 0.0314 | 0.0069 |

0.8500 | 0.0207 | 0.0298 | 0.0306 | 0.0064 |

0.9000 | 0.0205 | 0.0283 | 0.0291 | 0.0057 |

0.9500 | 0.0193 | 0.0255 | 0.0262 | 0.0041 |

1.0000 | 0.0157 | 0.0201 | 0.0205 | 0.0028 |

t | Exact Solution | Proposed Method | Method of Reference [29] |
---|---|---|---|

0 | 1.0000 | 1.0000 | 1.0000 |

0.1 | 1.1052 | 1.1051 | 1.1051 |

0.2 | 1.2214 | 1.2213 | 1.2213 |

0.3 | 1.3499 | 1.3498 | 1.3497 |

0.4 | 1.4918 | 1.4917 | 1.4917 |

0.5 | 1.6487 | 1.6486 | 1.6486 |

0.6 | 1.8221 | 1.8221 | 1.8220 |

0.7 | 2.0138 | 2.0137 | 2.0136 |

0.8 | 2.2255 | 2.2254 | 2.2253 |

0.9 | 2.4596 | 2.4595 | 2.4594 |

1 | 2.7183 | 2.7181 | 2.7181 |

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## Share and Cite

**MDPI and ACS Style**

Mosavi, A.; Shokri, M.; Mansor, Z.; Qasem, S.N.; Band, S.S.; Mohammadzadeh, A.
Machine Learning for Modeling the Singular Multi-Pantograph Equations. *Entropy* **2020**, *22*, 1041.
https://doi.org/10.3390/e22091041

**AMA Style**

Mosavi A, Shokri M, Mansor Z, Qasem SN, Band SS, Mohammadzadeh A.
Machine Learning for Modeling the Singular Multi-Pantograph Equations. *Entropy*. 2020; 22(9):1041.
https://doi.org/10.3390/e22091041

**Chicago/Turabian Style**

Mosavi, Amirhosein, Manouchehr Shokri, Zulkefli Mansor, Sultan Noman Qasem, Shahab S. Band, and Ardashir Mohammadzadeh.
2020. "Machine Learning for Modeling the Singular Multi-Pantograph Equations" *Entropy* 22, no. 9: 1041.
https://doi.org/10.3390/e22091041