Amel Serrat* | Bachir Djebbar
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
In this paper we study the aptitude of Artificial Bee Colony Programming (ABCP) to solve a Suspended Symmetric Drop Equilibrium Equation (SSDEE) described by a non linear second order differential equation. Artificial Bee Colony Programming was developed by Boudouaoui and Karaboga in 2020 to extend Artificial Bee Colony for symbolic regression problems as Genetic Programming (GP) where food source represents tree.The Suspended Symmetric Drop Equilibrium Equation is represented by a second order non linear Differential Equation with boundary conditions. The solving process is fully symbolic, and The Artificial Bee Colony Programming aims to find the best solutions according to fitness. We apply it with different parameters adjustment and we also apply the 4th Runge-Kutta method as usual method. The Artificial Bee Colony Programming solution was compared with the Fourth Runge-Kutta(4RK) and we found that Artificial Bee Colony Programming can solve Suspended Symmetric Drop Equilibrium Equation with a bite agreement and with fewer constraints.
artificial bee colony programming, differential equation, symbolic regression, suspended symmetric drop
Most real-world physical phenomena are modelled by differential equations and require solutions. Differential equations can be simple, complex or computationally time consuming [1-7]. The complexity of solution varies from one equation to another depending on whether it is ordinary or partial, linear or non-linear, with or without initial values or boundary conditions and depending on the order of the equation [3-7]. As a result, in this work, we present an artificial intelligence-based optimization and approximation technique for solving an ordinary nonlinear second order differential equation with boundary conditions.
The approach is known as Artificial Bee Colony Programming (ABCP), and it is based on Darwinian evolutionary principles such as 'natural selection and competition' [8-11]. The Artificial Bee Colony technique mimics the collective behavior of foraging bees [9, 11, 12] as they seek honey and pollen.
Many studies have increased the effectiveness and accuracy of the artificial bee colony method. Many changes in representation and design have been made to the Artificial Bee Colony Method, allowing it to function better. The performance of the artificial bee colony method has been demonstrated in a variety of applications involving high-complexity combinatorial problems in a variety of domains [13]. We can mention here image &video processing [14-18], data clustering [19-21], path planning [22], localization [23, 24], Scheduling [25-30], data-mining [31, 32] and symbolic regression [11, 33] and these are some of the fields.
The Artificial Bee Colony Programming technique, created by Karaboga and Ozturk in 2012 [9, 11, 12] to extend Artificial Bee Colonies, exhibits the same symbolic regression as Koza's Genetic Programming [34, 35]. The depiction of each individual in the latest version of The Artificial Bee Colony is represented by a tree [11]. Each individual in the foraging bee population corresponds to one of the problem's possible solutions, and the algorithm remains the same as that of Karaboga's artificial bee colonies developed in 2005 [11]. In 2020, Boudouaoui et al. [36] used Artificial Bee Colony Programming on some examples of differential equations and integrated the boundary conditions to the fitness function calculation [36-38].The constraint integration was constructed using the ordinary differential equation $f\left(x, y, y^{1}, \ldots, y^{n}\right)=0$ where fis a given function, .and yis an unknown function of x and the boundary conditions $g\left(x, y, y^{1}, \ldots, y^{n-1}\right)=0$ where gis a function that covers conditions:
$\begin{matrix} fitnes{{s}_{i}}=\sum\limits_{j}{f{{\left( {{x}_{j}},{{y}_{j}}\left( {{x}_{j}} \right),y{}_{j}^{1}\left( {{x}_{j}} \right),...,y_{j}^{i}\left( {{x}_{j}} \right) \right)}^{2}}} \\ +\lambda \sum\limits_{k}{abs\left( {{g}_{k}}\left( x,{{y}_{i}},y{}_{i}^{1},...,y_{i}^{n-1} \right) \right)} \\\end{matrix}$ (1)
With λ positive.
In this article, we propose using Artificial Bee Colony Programming to get a symbolic solution in some closed form to a Differential equation. The Artificial Bee Colony Programming try solve the equilibrium profile equation of a suspended drop on a solid wall, which is expressed as a nonlinear ordinary differential equation [37, 38]. Extensive trials are carried out, and the findings demonstrate that the approach is capable of achieving good performance.
The article is divided as follows: section 2 contains a description of the fundamental Artificial Bee Colony algorithm, and section 3 encloses a detailed discussion of the Artificial Bee Colony Programming method. Section 4 discusses the use of Artificial Bee Colony Programming and the findings achieved.
In this section, we introduce the associated work of our study, first presenting some related work regarding Differential Equation resolution.
We begin with analytical procedures [39] such as variable separation and integration, which yield reliable answers; and they try to find a solution in a closed form, where a finite number of elementary functions are used to express the solution. Nevertheless, these methods cannot solve all differential equations, and rewriting equations in a closed form takes time and requires a large mathematical background.
Numerical techniques [40] are an alternative to analytical methods which are simple to apply, yield an approximate solution, and are:
We can mention some of the numerical methods [40] like: finite difference, Runge Kutta, series expansion, divided difference and finite element and the Table 1 represents the most of them.
Optimization methods are another alternative to analytical methods like machine learning and evolutionary methods. We can mention:
Neural Net: it was the first technique [41], developed by Lee in 1989 as a model for the differential equation, is artificial neural networks. The applications are then based on error backpropagation. Shi and Xu [41] used it on several differential equations.
Deeplearning Neural Net: based on neural Net with complex training process and different learning method [42, 43]. The method was applied to high dimensional partial Differential equation [42], and in Ref. [43] to a system of differential equation with irregular solution.
Fuzzy Logic: based on fuzzy inference scheme [44]. It was applied to partial differential equation [44].
LS-SVM: based support Vector machine and least square method [45] was applied to high order ordinary differential equation.
Table 1. Numerical methods set
|
Finite Difference Method |
Runge Kutta Method |
Prediction & correction method |
|
Series Expansion Method |
Order Reduction Method |
Extrapolation method |
|
Finite Element |
Galerkin/ Bubnov Method |
Collocation Method |
|
Variable Step Method |
Shwarz Method |
Variable order Method |
Differential Evolution: based on population of candidate that optimizes a given problem [46, 47]. It was applied to Lorenz and Vander Pol differential equation, and to Partial Differential Equation.
Genetic Algorithm: based on evolutionary mechanism [48, 49]. It was applied to ordinary Differential Equation [48] and Delay differential equation [49].
Genetic Programming: based on Genetic Algorithm and Grammatical Evolution [50] in order to get a symbolic solution. It was applied [50] to several Differential Equation.
Ant Colony Programming: based on Artificial Ant Colony and Grammatical Evolution. It was applied Second Order Differential Equation [51].
Artificial Bee Colony is a method of optimization based on bee swarms. It was created by Karaboga [11, 12] to imitate the collective social behaviour of foraging bees while searching for pollen and nectar.
This behavior is governed by a single member of the bee colony, the worker. This worker performs several tasks during her life as in Figure 1.
We are only interested in two activities in this article: foraging (the exploration phase) and pollen and nectar collecting (the exploitation process), both of which are impacted by the following elements [37]:
Figure 1. Worker bee tasks
3.1 The basic behavior
The artificial bee colonies are made up of three sorts of worker bees: Scout, Employed, and Onlooker, and each type corresponds to a different processing phase in the algorithm.
The Scout bee searches for food sources X (potential solutions), and when discovered, it becomes an Employed bee.
The employed bee will compute the concentration of sugar and protein in a food source, the amount of nectar and pollen, and the location relative to the sun. In addition to the calculations, a sample of nectar and pollen is taken.
She returns to the hive after gathering this knowledge and shares it with her comrades, the Onlooker bees.
Employed bee performs a dance and allows the Onlooker bees to sniff the meal sample. The dance might be rapid or slow to notify them of the quantity, and the posture of the employed bee will indicate the location of the meal as in Figure 2 [11-33].
The onlooker bee watches the dance and detects the fragrance of the food before flying off to find and collect nectar and pollen [8-33].
The algorithm for artificial bee colonies is divided into three stages [11-33] as in Figure 3.
Figure 2. The Employed bee dance
Figure 3. The Ant Bee Colony flowchart
3.2 The basic operations in ABC method
The basic operations in Artificial Bee Colony Programming (ABCP) are: parameters initialization, tree construction, Fitness evaluation, information sharing, differentiation, and greedy selection [34-38]. We can report each operation and its requirements as follows:
Table 2. Terminals set
|
Terminals |
|
|
Constant |
0,1,2,3,4,5,6,7,8,9 |
|
Variables |
t |
Table 3. Functions set
|
Functions |
|
|
Arithmetic functions |
Sin, cos, exp, log, sqrt |
|
Operators |
+,-,*,/,(, ) |
3.2.1 Parameters initialization
The parameters are as follows [34-38]:
The Terminals collection. The terminal set corresponds to the content of the tree's leaves. The set of terminals is mention in Table 2.
It is made up of the variables and constants of the differential equation and is determined by the problem at hand as mention in Table 2.
The Functions collection. Except for the leaves, functions match to the content of the tree's nodes [34-38].
It is made up of mathematical functions such as addition, subtraction, division, multiplication, and other more complicated functions as in Table 3, and the set also varies depending on the issue being solved [34-38].
The Fitness function. The function assigns a value to each tree in the population. This procedure enables the algorithm to proceed with the individual selection (the strongest element of the population) [34-38].
The free parameters of control are:
3.2.2 Tree construction
Random tree construction [11] in ABC is a crucial operation in this method and can affect its performance. This operation is done by one of the following methods:
The Grow method. The leaves on trees [11, 52-55] created by the Grow technique are not all at the same depth (see Figure 4).
Figure 4. Tree created by the Grow method
The Full method. The process is similar to grow method with an exception that if the depth is less than the maximum allowable tree depth, a node is picked at random from a set of internal nodes rather than from the set of terminals. The leaves of the trees [11, 34-38, 52-53] created using the complete technique are all positioned at the same depth (see Figure 5).
Figure 5. Tree created by the Full method
The Ramped Half&Half. If the maximum depth of the tree [52-55] is N, the Ramped method will only allow for the creation of trees ranging in size from 02 to N, whereas the half and half method will ensure equity between the Grow method and the FULL method by generating 50% of the tree using the Grow method and the other half using the FULL method.
3.2.3 Fitness evaluation
The fitness value for each solution [11, 36-38] is calculated as in Eq. (2):
$f i t\left(X_{i}\right)=\left\{\begin{array}{c}\frac{1}{1+f\left(X_{i}\right)}, \text { si } f\left(X_{i}\right) \geq 0 \\ 1+a b s\left(f\left(X_{i}\right)\right), \text { else }\end{array}\right.$ (2)
where the evaluation of a solution is calculated as follows:
$f\left(X_{i}\right)=\frac{1}{2} \sum_{j=1}^{N}\left(\text { Desired }_{i}-\text { obtained }_{i}\right)^{2}$ (3)
3.2.4 Sharing information
The Onlooker and Employed bees [11, 36-38] exchange information about the quantity and quality of nectar and pollen $\left(X_{i}\right)$ in the symbolic manner (see Figure 6):
(1) Select randomly a solution $X_{k}$ a neighbor of $X_{i}$;
(2) Select randomly a solution $X_{k}$ a neighbor of $X_{i}$;
(3) Choose two nodes at random, the first on $X_{i}$ and the second on $X_{k}$, with a Coefficient $\beta$ to have a function and $\gamma$ to have a terminal;
(4) $V_{k}$ is obtained by replacing the first subtree on $X_{i}$ with the first subtree on $X_{k}$.
Figure 6. The Sharing information process
3.2.5 Differentiation
The ABC method guarantees [36-38] population variety by raising or lowering the odds during roulette selection by a probability.
This probability as in Eq. (4) is allocated [36-38] to each member of the population and is modified with each generation.
$P\left(X_{i}\right)=\beta *\left(\frac{f i t\left(X_{i}\right)}{\max (f i t)}\right)+\gamma$ (4)
3.2.6 Greedy selection
The aim of the Greedy selection [36-38] is to keep only the best and the choice is made as follows:
iffit $\left(V_{k}\right)>\operatorname{fit}\left(X_{i}\right)$ then $\left\{\begin{array}{c}\text { replace } X_{i} \text { by } V_{k} \\ \text { compt }_{i}=0\end{array}\right.$
else compt $_{i}=\operatorname{compt}_{i}+1$ (5)
3.3 The ABCP process
The Artificial Bee Colony Programming (ABCP) process uses three-phase: Bee-Employed, Bee-Onlooker, and Bee-Scout phases [11, 36-38]. These phases represent the various worker bee tasks involved in the search for a food source. We can report each phase and its requirements as follows:
3.3.1 Bee employed phase
Create a nearby solution [11] $V_{k}$ for each solution or tree $X_{i}$ of the population $X$ using the previously described information sharing method, compute its fitness, and then choose between $X_{i}$ and $V_{k}$ (see Figure 7).
Figure 7. The bee employed phase
3.3.2 Bee onlooker phase
Figure 8. The bee onlooker phase
The Bee Onlooker phase begins after a probability calculation for each individual in the population [36-38].
At random number generator [8] selects an individual $X_{i}$ from the population $X$, the information sharing mechanism creates an adjacent solution $V_{k}$, the fitness for this new individual $V_{k}$ is computed, and a choice between $X_{i}$ and $V_{k}$ is made (see Figure 8).
The Bee-Onlooker phase [11] can be seen as a single iteration of the Employed process where the individual $X_{i}$ is chosen by the roulette selection.
3.3.3 Bee scout phase
If a solution $X_{i}$ is not visited or updated for a certain period of time, it is considered abandoned and must be destroyed, [11] with a new one generated at random using the Grow technique.
The limit parameter formalizes the duration. Alternatively, if the variable compt approaches the limit, it will be reset to zero (see Figure 9).
Figure 9. The bee scout phase
The Artificial Bee Colony method is implemented to solve a second order non linear differential Equation.
Figure 10. A Suspended Drop on a solid wall
The equilibrium equation of the drop as in Eq. (6) describes the profile of a drop of Decane suspended as in Figure 10 from a Pyrex wall and dipped in salt water [51]:
$C_{1}-C_{2}+\frac{f^{2}(y)}{\left(1+f^{\prime}(y)^{2}\right)^{3 / 2}}=\frac{1}{f(y)\left(1+f^{\prime}(y)^{2}\right)^{1 / 2}}$ (6)
Under boundary conditions:
$\left\{\begin{array}{c}f(0)=x \\ f^{\prime}(0)=-\operatorname{cotg}(g \alpha)\end{array}\right.$ (7)
where,
$C_{1}=\frac{2}{b}+\frac{\Delta \rho g}{\sigma} h$ and $C_{2}=-\frac{\Delta \rho g}{\sigma}$ (8)
And the other drop parameters are mentioned in Table 4. The results of the Artificial Ant Colony programming method are compared with the Runge-Kutta method.
The Artificial Bee Colonies method is tested with different values of the following parameters: population size, the two degrees of visibility importance $\beta, \gamma$ as shown in Table 5.
Table 4. Drop profile parameters
|
Pyrex/Décane |
|
|
$\rho_{1}$ |
0.7268 |
|
$\rho_{2}$ |
1.0196 |
|
$\eta$ |
8.5 |
|
$V$ |
47.33 |
|
$\sigma$ |
60.53 |
|
$\theta_{1}, \theta_{2}$ |
56;56 |
Table 5. Comparison results of ABCP under different variatio
|
Parameters |
N° |
Value |
MSE |
Variance |
|
FN=PN |
01 |
30 |
2,400776E-03 |
1,107680E-03 |
|
02 |
40 |
2,234350E-03 |
2,353120E-03 |
|
|
03 |
50 |
2,726163E-02 |
2,044090E-03 |
|
|
04 |
60 |
1,774828E-03 |
2,073070E-04 |
|
|
05 |
70 |
2,424597E-03 |
1,479500E-04 |
|
|
06 |
80 |
2,004544E-02 |
1,828750E-03 |
|
|
07 |
100 |
2,375986E-03 |
2,570440E-04 |
|
|
08 |
150 |
1,611740E-04 |
2,054620E-05 |
|
|
09 |
200 |
1,816470E-03 |
2,703080E-04 |
|
|
10 |
300 |
1,772671E-03 |
2,410520E-04 |
|
|
11 |
400 |
2,533931E-03 |
3,496970E-04 |
|
|
$\boldsymbol{\beta}$ |
12 |
0.1 |
2,079761E-03 |
3,270530E-03 |
|
13 |
0.2 |
1,948729E-03 |
3,844950E-03 |
|
|
14 |
0.4 |
1,772671E-03 |
2,410520E-04 |
|
|
15 |
0.5 |
2,055930E-03 |
4,189800E-04 |
|
|
16 |
0.6 |
1,705610E-04 |
5,139720E-05 |
|
|
17 |
0.7 |
2,178042E-03 |
3,909670E-04 |
|
|
18 |
0.8 |
2,293738E-03 |
3,945580E-03 |
|
|
19 |
0.9 |
2,473200E-03 |
4,878640E-03 |
|
|
20 |
01 |
2,102692E-03 |
3,281240E-03 |
|
|
$\gamma$ |
21 |
0.1 |
1,72382012 |
4,37630023 |
|
22 |
0.2 |
2,73078764 |
3,59237056 |
|
|
23 |
0.4 |
2,202365E-01 |
5,225350E-01 |
|
|
24 |
0.5 |
1,998768E-03 |
6,159420E-01 |
|
|
25 |
0.6 |
2,649501E-03 |
5,950950E-03 |
|
|
26 |
0.7 |
1,127300E-04 |
4,770680E-03 |
|
|
27 |
0.8 |
2,305125E-02 |
5,686800E-05 |
|
|
28 |
0.9 |
1,772671E-03 |
2,410520E-04 |
|
|
29 |
01 |
1,816030E-02 |
8,345270E-01 |
The influence of $\beta, \gamma$ and population size PS (FS=PS) on optimization is reported in Table 5. In Table 5 we have compared different values on $\beta, \gamma$ and population size PS(FS=PS) with two metrics: Mean Squared Error (MSE) and variance.
The Mean Squared Error (MSE) is measured between the BCP output and the fourth Runge-Kutta results which is referred as the first mean order and variance as the second mean order. The best results are written in bold format.
Table 5 indicates that the suggested method based on Bee Colony Programming can achieve our aim in solving differential equations. That reveals the effectiveness and accuracy of soft commuting techniques, in the same manner as conventional methods.
This paper focuses on the efficacy of metaheuristic methods in solving differential equations.
To solve the equilibrium profile of a symmetric suspended drop as a nonlinear differential equation, the Artificial Bee Colony Programming method has been proposed.
We have proposed the Artificial Bee Colony Programming method as a variant of bee colonies but with a symbolic representation, analysis and optimization of individuals, the representation is in tree form and the optimization and analysis is applied on symbolic expressions.
The Artificial Bee Colony Programming method solves the differential equation representing the equilibrium profile of a symmetrical suspended drop as well as the Runge-Kutta method. The results show that the efficiency of the artificial bee colony programming method is comparable to the performance of the conventional Runge-Kutta method.
However, Artificial Bee Colony Programming still has a flaw in its solution search equation, which is strong at exploration but not so much at exploitation, we investigate an hybrid approach between Artificial Bee Colony Programming and Finite Element method to improve solution.
Mrs. Serrat Amel wishes to thank Pr. Djebbar Bachir for his guidance, suggestions, moral support, and patience in leading her through this endeavor.
|
abs |
Absolute value |
|
Fit |
Fitness value |
|
g |
Gravitational acceleration, m.s-2 |
|
P |
Probability value |
|
V |
Drop Volume, mm3 |
|
X |
Bee population |
|
Greek symbols |
|
|
$\alpha$ |
Contact Angle, ° |
|
$\beta$ |
thermal expansion coefficient, K-1 |
|
$\gamma$ |
solid volume fraction |
|
$\eta$ |
Distance x |
|
$\rho$ |
Fluid volume, g.cm-3 |
|
$\sigma$ |
Interfacial tension, dyne/cm |
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