Neural network modeling based double-population chaotic accelerated particle swarm optimization and diffusion theory for solubility prediction

Journal Pre-proof Neural network modeling based double-population chaotic accelerated particle swarm optimization and diffusion theory for solubility prediction Mengshan Li, Suyun Lian, Fan Wang, Yanying Zhou, Bingsheng Chen, Lixin Guan, Yan Wu

PII:

S0263-8762(20)30004-6

DOI:

https://doi.org/10.1016/j.cherd.2020.01.003

Reference:

CHERD 3953

To appear in:

Chemical Engineering Research and Design

Received Date:

29 July 2019

Revised Date:

2 December 2019

Accepted Date:

2 January 2020

Please cite this article as: Li M, Lian S, Wang F, Zhou Y, Chen B, Guan L, Wu Y, Neural network modeling based double-population chaotic accelerated particle swarm optimization and diffusion theory for solubility prediction, Chemical Engineering Research and Design (2020), doi: https://doi.org/10.1016/j.cherd.2020.01.003

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Neural Network modeling based double-population chaotic accelerated particle swarm optimization and diffusion theory for

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Solubility prediction

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Mengshan LI*, Suyun LIAN, Fan WANG, Yanying ZHOU, Bingsheng CHEN, Lixin GUAN, Yan WU

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College of Physics and Electronic Information, Gannan Normal University, Ganzhou, 341000, Jiangxi,

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CHINA;

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Correspondence to: Mengshan LI (E-mail: [email protected])

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Graphical abstract

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Highlights

A novel method is proposed for solubility prediction.



An improved PSO algorithm based on the diffusion theory is proposed.



RBF ANN trained by improved APSO is proposed.



Proposed method has a broad application perspective

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

ABSTRACT

Solubility is as a key chemical and physical property. Solubility prediction methods are applied in diverse fields including preparation synthesis and modifications of 2

materials. To overcome the shortcomings of existing solubility prediction methods, taking the mass transfer of two-phase system as an example, a solubility prediction model based on the diffusion theory and hybrid artificial intelligence method was proposed in this paper. An improved double-population chaotic

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accelerated particle swarm optimization (APSO) algorithm combined diffusion theory was developed according to the particle evolution utilizing diffusion energy.

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The developed algorithm was applied in the training of parameters of the radial

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basis function artificial neural network and then a model for predicting solubility

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was developed. The experimental results of supercritical carbon dioxide solubility in 8 polymers were consistent with the predicted values by the model, indicating

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the high prediction accuracy. The average relative deviation, squared correlation

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coefficient, and root mean square error were respectively 0.0036, 0.9970, and 0.0152, displaying its higher comprehensive performance. The model may also be

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applied in other physicochemical fields.

ABBREVIATIONS

PVT

Pressure, Volume, Temperature

ANN

Artificial Neural Network 3

RBF

Radial Basis Function

PSO

Particle Swarm Optimization

CAPSO

Chaotic accelerated PSO

Poly(butylene succinate).

PBSA

Poly(butylene succinate-co-adipate).

PS

Polystyrene.

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Polypropylene.

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PP

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PBS

Carboxylated polyesters

PLLA

Poly(l-lactide)

Poly(D,l-lactide-co-glycolide) High-density polyethylene

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HDPE

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CPEs

PLGA

ARD R2

MSE

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DP-DT-CAPSO Double population CAPSO based on diffusion theory

Average Relative Deviation

Squared Correlation Coefficient Mean Square Error 4

RMSEP

Root Mean Square Error of Prediction

Keywords: Neural Network; double-population; particle swarm optimization;

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diffusion theory; Solubility prediction

1. Introduction

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Solubility of supercritical carbon dioxide (SCCO2) in polymeric compounds is

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important for the modifications, synthesis, and preparation of new materials[1-6].

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Under high-temperature and high-pressure supercritical conditions, a solubility experiment is time-consuming and costly and it is not easy to obtain experimental

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data. Therefore, it is necessary to design a prediction model with high accuracy. The solubility of SCCO2 in polymers is affected by polarity of molecules, density, temperature, and pressure. These factors show the complicated non-linear

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associations with the solubility and the relationships among these factors are also complex. Therefore, traditional prediction methods based on thermodynamic equation of state or other empirical equations cannot provide the satisfactory prediction accuracy[7-12]. Ziaee et al.[13] predicted carbon dioxide solubility in different polymers with support vector machine and indicated that the proposed 5

models were efficient. Artificial neural network (ANN) possesses the ability of self-organizing, non-linear processing, and fault tolerance and can overcome the challenges of prediction[14-23]. Bakhbakhi et al.[24] and Lashkarbolooki et al.[25] compared the performances of ANN and the thermodynamic equation of

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state in solubility prediction and showed that the ANN method possessed the higher prediction accuracy than the thermodynamic equation of state.

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Gharagheizi [26] showed that the ANN model had a higher prediction accuracy of

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the SCCO2 solubility in compounds. Similarly, through SCCO2 solubility prediction

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experiments, Eslamimanesh et al.[27] demonstrated a high performance of ANN. Pahlavanzadeh et al.[28] predicted the solubility based on ANN model and

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Deshmukh-Mather method and demonstrated that ANN model was more

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effective than conventional thermodynamic equations. The accuracy and reliability of ANN are dependent on the adopted training algorithm and the essence of training is to optimize the model structure and

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variables. Several algorithms have been developed to optimize ANN training, including ant colony algorithm[29, 30], particle swarm optimization algorithm (PSO)[31-33], simulated annealing algorithm, and genetic algorithm[34]. With the online strategy and PSO algorithm, Liu et al.[35] trained the fuzzy neural network 6

and made a successful prediction of the melt flow rate (MFR). Lazzus et al.[36] used PSO algorithm to predict the phase equilibrium data of SCCO2 and achieved the better prediction results. Khajeh et al.[37] proposed the solubility prediction method of ANN and ANFIS and realized the higher prediction accuracy compared combined Kent-Eisenberg model with

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to traditional methods. Hussain et al.[38]

ANN, proposed a hybrid neural network solution model, and achieved the better

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performance in terms of prediction. LI et al. [39-45] combined the chaos theory

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with the PSO and clustering methods, proposed several solubility prediction

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models through improving ANN training algorithm, and achieved the higher

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prediction accuracy.

Previous studies mainly focused on improving the model itself without

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considering the solubility element. Under static conditions, the solution process is basically a diffusion process in which solute molecules are adsorbed onto the surface. This process is driven by thermal movement of molecules and dependent

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on the gradient of temperature or density between two media. Solute molecules migrate from the denser medium to the less dense medium continuously to produce an equilibrium at which the movement of solute molecules stops. In a two-phase system of polymer/SCCO2, at the interfacial film layer, due to interface 7

force, CO2 molecules attached to the interface spread to the interfacial film and are dissolved in the melted polymer. PSO algorithm may be used to analyze the solution process. Firstly, particles in PSO algorithm resemble those in solution and the similarity can be simulated by assigning the associated properties of solute

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molecules in solution to the particles of the algorithm. Secondly, the solution process is a mass transfer process caused by the transport of material and it is

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closely related to the diffusion theory. Based on the diffusion theory and solubility

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and particle evolution algorithms, in order to simulate the molecular diffusion

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process, we explored the movement of CO2 in the diffusion process and simulated the diffusion process based on the movement of evolution particles. Currently,

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the PSO algorithm was seldom used to analyze the migration of particles,

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particularly in thermodynamics. Therefore, in this investigation, the diffusion theory was introduced into the PSO algorithm to improve it. The solution process of SCCO2 in polymer was simulated with the PSO algorithm and then an improved

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PSO algorithm was proposed combined the diffusion theory and then applied in ANN training. Finally, a hybrid artificial intelligence prediction model of SCCO2 solubility in polymers was obtained.

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2. Model theory 2.1. Standard PSO algorithm In 1995, Eberhart and Kennedy described the particle swarm optimization (PSO) as an evolution algorithm[46]. The movement of the particle is illustrated by the

k

k

k

k

(1)

k

v i, d =  v i, d + c 1 (p i, d - x i, d ) + c 2 (p g, d - x i, d )

k +1 i, d

k

k +1

i, d

i, d

= x +v

(2)

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x

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k +1

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following formulas:

k

p i,d

and

k

p g,d

and

k

x i,d

are respectively the velocity and position of

are the position of the individual and global extrema.

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particle i;

k

v i,d

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are learning factors;

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where i = 1, …… , m (m indicates particle number);  is inertia weight; C1 and C2

2.2. CAPSO Algorithm

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In recent years, an improved PSO variant with the extremely high global convergence performance, accelerated particle swarm optimization (APSO), has been widely concerned among scholars. In the APSO algorithm, inertial weight factors or cognitive factors are not considered and global exploration factors are utilized to improve the algorithm. The algorithm considers the global search 9

exploration factors for particle update. In the entire search process, particles are only restrained by global extremum, so the search velocity is accelerated. Position update formula is provided as: k +1

k

k

x i, d = (1 - C 2 )x i, d + C 2 pg, d

(3)

 C1 r

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where C1r is a random number and can allow the algorithm to escape from the

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local optimum.

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Compared to the standard PSO algorithm, APSO adopts two variables C1 and C2 to

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reduce randomness during iteration. C1 is expressed as a monotonically decreasing function: C1   t , 0    1 . Thus, in the APSO algorithm, C2 is the key

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determinant of the algorithm performance and its range is [0.2, 0.7]. When C2 is 1,

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the particles can converge to the current global extreme at any time and show no change and this global extremum barely represents the actual global extremum. Conversely, when C2 is 0, the search velocity of the algorithm is extremely slow.

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Hence, it is important to optimize C2 by analyzing the performance of APSO algorithm.

However, in the APSO algorithm, the premature convergence problem still exists and some extrema may be avoided. According to the characteristics of learning 10

factor C2 in APSO, it can be described by chaotic mapping, and chaos theory can be used to optimize the parameter C2. Therefore, the classical logistic equation can be applied to achieve evolution and optimization of chaotic parameters. The used iterative formula is provided as follows: k 1 i

= 4x (1  x ) k

k

i

i

(4)

k

0 < xi  1

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x

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2.3. Double-population CAPSO algorithm with diffusion theory

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In the application of CAPSO, a particle is designed as a potential solution to the actual problem and the optimal solution is updated by the iteration of the

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algorithm. There are two phases in the system: SCCO2 melt and polymer melt. The

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molecular energy required to overcome the barrier from the original position to other locations, which is called the diffusion energy. The molecules move with the

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diffusion energy, and then achieve mass transfer. The diffusion velocity of a molecule is proportional to the movement speed and the temperature of the system determines the movement speed of the molecule. In the molecular

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system, the system temperature is a statistical variable, which is also proportional to the average kinetic energy of the molecule. Temperature is proportional to the speed of molecular movement.

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Based on the thermodynamic diffusion theory, a novel CAPSO algorithm, called DP-DT-CAPSO algorithm, was proposed in this paper. In the DP-DT-CAPSO algorithm, the molecules movement is replaced by the particles movement to simulate the molecular force field on particles. The diffusion temperature is

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determined by the thermal motion of the molecules. The algorithm assumes that the particle has a population temperature, which represents the average

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molecular temperature of the system. The related concepts and definitions are

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respectively described below.

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Definition 1: Particle diffusion energy

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2.3.1. Related concepts and definitions

1 E  mv 2 2

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In CAPSO algorithm, the kinetic energy of the particle population is defined as: (5)

where v and m are respectively the speed and mass of the particles. The particle

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with kinetic energy can spread to a new position when it can overcome the movement barrier and the kinetic energy is called the particle diffusion energy. If the mass of all particles is 1, the particle diffusion energy

Qi 

1 n 2 Vij 2 j 1

(6) 12

Qi

can be defined as:

where

is the speed of the particles.

Vij

Definition 2: Center of mass of the population The convergence probability of PSO algorithm is 100% when all the particles move toward a center. Therefore, in the entire population, a population center is set

x m m i

(7)

i

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X cen 

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and the particles move toward the center, the formula is updated as follows:

where

X cen

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i

is the center of mass of the population; m is the size of the

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population ; xi is the position.

Definition 3: Distance to center of mass di

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The distance to center of mass di  xi

Xc

is expressed as: (8)

e n

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Definition 4: Particle diffusion probability The particle diffusion probability indicates the proportion of successful diffusion of particles that meet the diffusion conditions and is depending on both energy and temperature. Particle diffusion probability 13

Pi

is defined as:



Qi

1 e T Pi  d

(9)

where T is temperature.

d

is the distance difference between the particles to

the centroids. (10)

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B d  xi  X cenA  xi  X cen

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2.3.2. DP-DT-CAPSO algorithm

The SCCO2/polymer solution system, which have two-phase, one is the gas phase

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and the other is the melt phase. Therefore, we used two populations to model

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the two-phase system. Population A is the gas molecular population and

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Population B is the polymer melt molecular population. The operation and function of the two phases are basically the same in the algorithm.

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Taking Population A as an example, in the execution of the DP-DT-CAPSO algorithm, the particles diffusion energy in Population A is first calculated and then the particles diffusion probability is obtained. The particles with higher

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diffusion probability are copied to the candidate Population A, the particle carried the largest diffusion probability is used to update the worst fitness particles in the candidate Population B. Then, the global extremum is obtained. The algorithm is executed as follows: 14

Step 1: Initialization. Population A and Population B are initialized and then the maximum number of iterations are set. The values of some parameters such as population size, particle fitness value, and particle position are randomly set. Step 2: Fitness value calculation. The fitness value of the population particles is

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calculated.

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Step 3: Extremum value update. According to the fitness value of the particle, the

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extremum of the individual particle in the population is updated.

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Step 4: Termination condition calculation. When the number of iterations reaches the maximum value or the convergence accuracy meets the criteria, go to Step 11;

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otherwise, go to the next Step.

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Step 5: the diffusion probability of population particles is calculated. Step 6: Candidate population. If the particles diffusion probability is greater than the generated random number, the particles are copied into the candidate

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population.

Step 7: Information sharing and diffusion between population particles. Taking Population A as an example, the particles with the highest diffusion probability are selected from A and the particles with the worst fitness values among the 15

Candidate Population B are replaced. After particle information sharing, the mass transfer is completed. Step 8: Update the global extremum. When the particle's fitness value is better, the global extremum is updated to the current particle.

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Step 9: Iterative process data are saved. Various groups and candidate sets and

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extreme values are saved.

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Step 10: Iterative update. Particle attributes and population attributes are

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Step 11: The final result is saved.

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updated.

2.4. Artificial neural network trained by the DP-DT-CAPSO

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Among many artificial neural networks, RBF ANN has unique advantages and is one of the widely concerned models. It has a three-layer structure. In this paper, Gaussian function was selected as the activation function: xk  ci

Jo gi ( xk )  exp(

 i2

2

(11)

)

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where xk (1  k  n) is the output vector of the K-th iteration; ci (1  i  c) is the center of the basis function;  i is the expansion factor; n is the number of samples. In the training process of RBF ANN network, three parameters are mainly

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optimized: the basis function center, the extension constant, and the connection

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weight. The network is trained through the iterations of ci ,  i , wi . The output of

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RBF ANN network is defined as: c

O( xk )   wi gi ( xk )

(12)

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i 1

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where wi is the connection weight of the i-th hidden node. In this paper, we used the DP-DT-CAPSO algorithm to train the three parameters

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ci ,  i , wi The structure of the particle is defined as: . y  f (wh,o ,  h,o , ci )

(13)

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where wh,o (1  h  c) (1  o  p) is the weight;  h,o is the extension factor between the hidden node and output node.

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A prediction model based on RBF ANN and DP-DT-CAPSO algorithm applied in the prediction of SCCO2 solubility in polymers is developed, called DP-DT-CAPSO RBF ANN. 3. Model establishment

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3.1. Experimental data

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Firstly, 327 sets of data of 8 common polymers in industrial production are used

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to establish the model database and the data sources are provided in Table 1.

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In order to obtain the better performance, in this paper, we divided the experimental data into three subsets randomly. The training set (including 70% of

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the data) was used for the model training. The verification set (including 15% of the data) was used to optimize the trained model. The testing set (containing 15%

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of the data) was used to test the the model. Table 2 shows the data distribution statistics of each polymer. 3.2. Model evaluation

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The average relative deviation (ARD), root mean square error of prediction (RMSEP), and squared correlation coefficient (R2) were used to evaluate the performance of the model in this paper.

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ARD 

1 N y i  yi  N i 1 yi

RMSEP 

1 N

(14)

N

 ( yi  yi )2

(15)

i=1

2

N    ( yi  yave )( yi  yave )   R 2  N i=1 N  ( yi  yave )2  ( yi  yave )2

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i=1

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i=1

(16)

yi

and yave are the experimental value and

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and the predicted average value;

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where N is the sample number; yi and yave are respectively the predicted value

3.3. Model structure

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the average data of experimental value.

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The proposed model uses a three-layer structure composed of the input layer, the hidden layer and the output layer. The input layer contains two nodes of temperature and pressure. The output layer consists of one node indicating the

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solubility of SCCO2 in the polymer. In order to see the effect of the nodes in the hidden layer on the performance of the model, the number of nodes in the hidden layer is optimized by the heuristic method. The number of nodes is changed from 3 to 13. Therefore, 11 models are obtained. Fig. 1 shows the error curves with different nodes in the hidden layers. 19

As shown in Fig. 1, as the number of nodes increases, the error firstly decreases and then increases. When the number of nodes in the hidden layer is 6, the error is the smallest. Therefore, in this paper, the number of nodes in the hidden layer is 6.

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4. Results and discussion

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4.1. Results of the proposed model

Experiments were performed in Windows 7 SP1 64-bit OS (4.00 GB of memory

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and Intel (R) Core ™ i5-4460 processor). Through Matlab 2010a software

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programing, a three-layer model with the 2-6-1 structure, called DP-DT-CAPSO

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RBF ANN, was developed. The proposed model was used to calculate the SSCO2 solubility in 8 polymers. In the training set, Fig. 2 draws the experimental values

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and the predicted values of the proposed model. As shown in Fig. 2, in the training set, the predicted data of 8 polymers are close to the experimental data and the distribution of the predicted data is similar to

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the ideal prediction line. The predicted values were consistent with the experimental values, indicating that the DP-DT-CAPSO RBF ANN model was well trained. Fig. 3 shows the data distribution of the predicted and experimental values in the validation set. The prediction data of the model were close to the 20

experimental data (Fig. 3). The vertical distance from the data point to the straight line indicated that the DP-DT-CAPSO RBF ANN model had the higher prediction accuracy, and had a better correlation. The distribution of the predicted data of the model was close to that of the

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experimental data (Fig. 4), indicating that the prediction performance was good.

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The prediction points in the testing set were relatively more dispersed than

others, indicating that the accuracy in the testing set was slightly lower. Table 3

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shows the model performance values for each data set.

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It can be seen from Table 3 that the model has a better prediction performance,

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higher accuracy and correlation in the three sets. Table 4 shows the related data of the model of predicting the SCCO2 solubility in 8 kinds of polymers. The data

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showed that the prediction accuracy of the model was basically the same among 8 polymers. The correlation coefficient was also above 0.99. The model showed

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the better prediction ability and better scalability. 4.2. Comparison with other models The above confirmed that the higher performance of the proposed model in predicting the SCCO2 solubility in polymers. In order to further explore the performance, it is necessary to carry out the comparison between the current 21

work and previous works, especially those ones based on the artificial intelligence and ANN. Therefore, the model proposed in this paper was compared with three types of models used PSO and RBF ANN, such as RBF ANN, PSO BP ANN PSO RBF ANN. The convergence curves of these compared models are shown in Fig. 5.

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As shown in Fig.5, the models of DP-DT-CAPSO RBF ANN, PSO RBF ANN, PSO BP

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ANN and RBF ANN tended to be stable after 200, 400, 300 and 700 times,

respectively. The model proposed in this paper has faster convergence speed. The

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prediction accuracy of the models decreased as the following order: DP-DT-CAPSO

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RBF ANN > PSO RBF ANN > PSO BP ANN > RBF ANN. The accuracy of the

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DP-DT-CAPSO RBF ANN model was close to 0. The proposed model is better in terms of convergence accuracy and convergence speed. Fig. 6 shows the

model.

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correlation between predicted and experimental values for each comparison

As shown in Fig. 6, various models have different prediction data distributions and

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the prediction values of the proposed model are closer to the experimental data. In terms of prediction accuracy, the models decreased as the following order: DP-DT-CAPSO RBF ANN > PSO RBF ANN > PSO BP ANN > RBF ANN. The

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DP-DT-CAPSO RBF ANN model had the highest prediction accuracy. Table 5 provides the statistical parameters of each model. In terms of the ARD and RMSEP data, the prediction accuracy of DP-DT-CAPSO RBF ANN model was better. In term of the correlation coefficient, the

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DP-DT-CAPSO RBF ANN model was also significantly better than other models.

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The above data indicated that the proposed model exhibited the better

performance in terms of accuracy and correlation. Table 6 gives the computation

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time of various models.

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Execution time of DP-DT-CAPSO RBF ANN model was close to that of RBF ANN

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model and shorter than that of PSO BP ANN and PSO RBF ANN. In the intelligent algorithm-based model, the introduction of the intelligent algorithm increased

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calculation time. The diffusion theory was introduced in the particle swarm optimization algorithm in this paper. The thermodynamic parameters were recalculated in the iteration, thus resulting in longer calculation time of the model.

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Compared with other models, this model also showed acceptable calculation time.

4.3. Discussion 4.3.1. Correlation of the model 23

The above experiments confirmed the advantages of the DP-DT-CAPSO RBF ANN model in terms of calculation accuracy, correlation and calculation time. Numerous experiments showed that solubility was proportional to the pressure of the system and inversely proportional to temperature. Fig. 7 shows the solubility

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curves of the proposed model under different temperatures and pressures.

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The solubility of SCCO2 in polymers (PP, HDPE, CPEs, PS, PBS and PBSA) increased with the increase in pressure and decreased with the increase in temperature (Fig.

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7). The solubility of SCCO2 in polymers (PLLA and PLGA) increased with the

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increase in pressure. With the increase in temperature, the solubility SCCO2 in

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PLLA and PLGA decreased firstly and then became stable. The predicted trend was consistent with the experimental results.

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4.3.2. Stability of the model

According to Table 4 and Table 5, the proposed model has the better prediction ability of SCCO2 solubility in the 8 polymers, displaying the higher accuracy and

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correlation. There is no abnormal prediction point and the model is the better stability. This paper proposes a model by combining thermodynamic diffusion theory with artificial intelligence, such as swarm intelligence algorithms. Compared with previous studies, it has several obvious characteristics. 1) In terms 24

of prediction accuracy and correlation, Table 4 and Table 5 show that the model in this paper has obvious advantages. 2) Empirical correlation exists in all prediction models based on artificial intelligence algorithms. In this paper, through the separation of the training set, verification set and test set, the

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interference of empirical correlation is minimized. 3) From the efficiency of the intelligent model, according to the calculation time in Table 6, it can be seen that

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the efficiency of various models based on intelligent algorithms is better.

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4.3.3. Fault tolerance and the anti-jamming ability of the model

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After the model is trained with many normal data, invalid data or abnormal data

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can be avoided by the model. The trained model displays the better fault tolerance and strong anti-jamming ability.

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4.3.4. Expansibility of the model

Solubility prediction experiments confirmed the higher prediction performance of the model. In addition, the model can be extended to other applications such as

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the prediction of other chemical and chemical properties, such as data prediction, data processing and optimization of experimental parameters, quantitative structure-activity relationship, and computer-aided design. The model has good scalability. 25

5. Conclusions In this paper, an improved model was proposed based on diffusion theory, particle swarm optimization and artificial neural network and used to predict the solubility of SCCO2 in 8 polymers. Experimental results showed that the

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DP-DT-CAPSO-RBF-ANN model was better than other comparison models in terms of prediction accuracy, correlation and calculation time. The model has good

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stability and scalability and can be applied in prediction, fitting, and processing of

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various types of the data of chemical, physical, biological, and information fields.

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Declaration of interests

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methods and prediction models.

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In the future, we will further explore more efficient theoretical calculation

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant Numbers: 51663001).

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36

of ro -p re

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ur na

mean squared error

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Fig. 1. Correlations between the number of nodes of in the hidden layer and

37

of ro -p

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ur na

lP

re

Fig. 2 Experimental values and predicted values in the training set.

Fig. 3 Experimental values and predicted values in the validation set.

38

of ro

-p

Fig. 4. Relationship between predicted values and experimental data in the

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ur na

lP

re

testing set.

Fig. 5 Curve of mean squared error versus iteration number

39

of ro -p

Jo

ur na

lP

re

Fig. 6 Predicted values VS. experimental data.

40

of ro -p re lP ur na Jo

Fig. 7. Solubility variations with pressure and temperature

41

Table 1 Sources and statistics of experimental data in this work T (K)

P (MPa)

Solubility(g/g）

Data points

References

PP

313.20-483.70

7.400-24.910

0.03950-0.26170

67

[47-50]

PLLA

308.00-323.00

9.620-31.460

0.16520-0.43010

27

[51]

HDPE

433.15-473.20

10.731-18.123

0.00551-0.12296

20

[47, 48]

CPEs

306.00-344.00

10.150-31.020

0.09840-0.63660

56

[52]

PS

338.22-473.15

7.540-44.410

0.02641-0.16056

70

[47, 53-55]

PBS

323.15-453.15

8.008-20.144

0.04534-0.17610

31

PBSA

323.15-453.16

7.870-20.128

0.04763-0.17411

29

PLGA

308.00-323.00

10.140-31.470

0.09030-0.29630

27

Total

306.00-483.70

7.400-44.410

0.00551-0.63660

327

PP

47

PLLA

19

HDPE

14

CPEs PS PBS PBSA

ro

[47, 56]

-p

[51]

Testing

Total

10

10

67

4

4

27

3

3

20

38

9

9

56

50

10

10

70

21

5

5

31

21

4

4

29

19

4

4

27

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PLGA

Validation

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Training

[47, 56]

ur na

Polymer

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Table 2 Distribution of experimental data

of

Polymer

42

Table 3 Values of ARD, R2, and RMSEP for different subsets. ARD

R2

RMSEP

Training

0.0036

0.9973

0.0147

Validation

0.0035

0.9971

0.0152

Testing

0.0037

0.9966

0.0157

Average

0.0036

0.9970

0.0152

of

Subset

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Table 4 Values of ARD, R2, and RMSEP for different polymers in the testing set. Polymer

ARD

R2

PP

0.0031

0.9972

PLLA

0.0037

0.9966

0.0138

HDPE

0.0042

0.9967

0.0137

CPEs

0.0037

PS

0.0037

PBS

re

-p

RMSEP 0.0147

0.0152

0.9964

0.0146

0.0042

0.9967

0.0146

PBSA

0.0036

0.9966

0.0152

PLGA

0.0033

0.9968

0.0146

Jo

ur na

lP

0.9973

43

Table 5 Statistical parameters of various models CPEs

PS

PBS

PBSA

PLGA

Average

RBF ANN

0.0112

0.0096

0.0113

0.0104

0.0106

0.0121

0.0111

0.0098

0.0108

PSO BP ANN

0.0078

0.0076

0.0086

0.0081

0.0068

0.0089

0.0088

0.0075

0.0080

PSO RBF ANN

0.0065

0.0064

0.0072

0.0056

0.0064

0.0078

0.0076

0.0063

0.0067

DP-DT-CAPSO RBF ANN

0.0023

0.0031

0.0042

0.0037

0.0034

0.0041

0.0038

0.0031

0.0035

RBF ANN

0.9589

0.9613

0.9621

0.9558

0.9642

0.9616

0.9643

0.9621

0.9613

PSO BP ANN

0.9853

0.9842

0.9823

0.9813

0.9811

0.9911

0.9813

0.9821

0.9836

PSO RBF ANN

0.9875

0.9876

0.9871

0.9856

0.9845

0.9842

0.9852

0.9834

0.9856

DP-DT-CAPSO RBF ANN

0.9972

0.9963

0.9966

0.9971

RBF ANN

0.0664

0.0589

0.0687

0.0921

PSO BP ANN

0.0482

0.0472

0.0399

PSO RBF ANN

0.0421

0.0423

0.0421

DP-DT-CAPSO RBF ANN

0.0149

0.0133

0.0137

ro

of

HDPE

0.9968

0.9962

0.9966

0.9967

0.9967

0.0887

0.0961

0.0952

0.0872

0.0817

-p

RMSEP

PLLA

0.0467

0.0466

0.0512

0.0514

0.0488

0.0475

0.0442

0.0475

0.0468

0.0476

0.0472

0.0450

0.0142

0.0146

0.0149

0.0147

0.0144

re

R2

PP

lP

ARD

Model

0.0145

Table 6 Computation time of various models

RBF ANN PSO BP ANN PSO RBF ANN

ur na

Model

Computation Time(S) 32 41 43 27

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DP-DT-CAPSO RBF ANN

44