Modeling the CO2 separation capability of poly(4-methyl-1-pentane) membrane modified with different nanoparticles by artificial neural networks

Membranes are a potential technology to reduce energy consumption as well as environmental challenges considering the separation processes. A new class of this technology, namely mixed matrix membrane (MMM) can be fabricated by dispersing solid substances in a polymeric medium. In this way, the poly(4-methyl-1-pentene)-based MMMs have attracted great attention to capturing carbon dioxide (CO2), which is an environmental pollutant with a greenhouse effect. The CO2 permeability in different MMMs constituted of poly(4-methyl-1-pentene) (PMP) and nanoparticles was comprehensively analyzed from the experimental point of view. In addition, a straightforward mathematical model is necessary to compute the CO2 permeability before constructing the related PMP-based separation process. Hence, the current study employs multilayer perceptron artificial neural networks (MLP-ANN) to relate the CO2 permeability in PMP/nanoparticle MMMs to the membrane composition (additive type and dose) and pressure. Accordingly, the effect of these independent variables on CO2 permeability in PMP-based membranes is explored using multiple linear regression analysis. It was figured out that the CO2 permeability has a direct relationship with all independent variables, while the nanoparticle dose is the strongest one. The MLP-ANN structural features have efficiently demonstrated an appealing potential to achieve the highest accurate prediction for CO2 permeability. A two-layer MLP-ANN with the 3-8-1 topology trained by the Bayesian regulation algorithm is identified as the best model for the considered problem. This model simulates 112 experimentally measured CO2 permeability in PMP/ZnO, PMP/Al2O3, PMP/TiO2, and PMP/TiO2-NT with an excellent absolute average relative deviation (AARD) of lower than 5.5%, mean absolute error (MAE) of 6.87 and correlation coefficient (R) of higher than 0.99470. It was found that the mixed matrix membrane constituted of PMP and TiO2-NT (functionalized nanotube with titanium dioxide) is the best medium for CO2 separation.


Gathered data from the literature
As already discussed, permeability is one of the key specifications of membrane technology for gas separation, which is often experimentally measured. On the other hand, several other studies have investigated the impact of employing different nanoparticles to improve the performance of polymetric membranes to this end. Accordingly, this study has developed a robust theoretical topology to estimate the CO 2 permeability in the pure PMP and PMP/nanoparticle mixed matrix membranes, which to the best of the authors' knowledge is the first one in this area. In this way, the nanoparticle types, their weight percentage (wt%) in the fabricated membrane, and operating pressure are the independent variables to estimate the CO 2 permeability in a specific membrane. Table 1 presents the main statistical features of the gathered experimental data from the literature [58][59][60][61] .
It is noteworthy that the literature has added up to 40 wt% of four nanoparticles (i.e., TiO 2 , ZnO, Al 2 O 3 , and TiO 2 -NT) to the PMP structure to fabricate different mixed matrix membranes. Also, 112 CO 2 permeability tests have been conducted in a pressure range of 2-25 bar. The CO 2 permeability of 18.01-570.90 barrer was Table 1. Literature data for the CO 2 separation by the PMP-nanoparticle membranes 58 [58][59][60][61] . Since this study includes both qualitative (additive type) and quantitative (nanoparticle dose and pressure) independent variables, it is also necessary to represent the earlier quantitatively. Table 2 introduces the numerical codes used in this regard.
Histograms of all independent (additive type, nanoparticle dose, and pressure) and dependent (CO 2 permeability) variables are depicted in Fig. 1.

Artificial neural networks
Artificial neural networks (ANNs) as a biologically inspired computational approach is a non-linear topology, which has a high capacity for data processing in the engineering area 62 . Actually, the ANNs are a reduced set of concepts derived from biological neural systems based on the simulation of data processing of the human brain and nervous systems 63 . The ANNs have already proved a robust potential for statistical analysis in the area without a broad range of experimental values regarding their flexibility and capability 62,63 . In the way of deriving an ANN paradigm, it is required to specify the main independent variables that affect the output of the process. It is worth noting that the ANNs have the potential to correlate the dependent variables with the independent ones with any degree of complexity 64 . To this end, providing a proper dataset is necessary to design a black box for the estimation of dependent factors considering defined criteria 62 . Accordingly, the obtained approach develops a signal among the input and output factors, which specifies the details in different layers related to neuron interactions.
Up to date, several ANN approaches have been developed, including multi-layer perceptron (MLP-ANN) 65 , radial basis function (RBF-ANN) 66 , cascade feedforward (CFF-ANN) 67 , general regression (GR-ANN) 68 , which the MLP-ANN is the most commonly used one. Generally, the MLP-ANN is an online learning supervised procedure that employs partial fit order together with tunable synaptic weights 69 . On these grounds, this topology was applied in this work to estimate the permeability of CH 4 and N 2 in PMPs. Routinely, an MLP-ANN is developed by defining three main layers, including the input layer, the hidden layer, and the output one. In this way, the input layer is derived from the raw independent (input) values after some data processing, which has  www.nature.com/scientificreports/ already proven their high impact on the process. Then, the outcome of this layer is introduced to the hidden layer to employ statistical analysis and mathematical treatment on the data. Afterward, the outcomes of this layer are transferred to the output layer that specifies the main results of the model. It should be considered that the major mathematical processing employed on the neurons is determined by Eq. (1) 70 : here b specifies the bias of the model, which indicates the activation thresholds for input values ( x r ), and ω jr is the weight coefficients of the model. Also, the net output of neurons ( O j ) is received by a transfer function ( tf ) to calculate the neuron's output 70 . In this work, the hyperbolic tangent sigmoid (Eq. 2) and logarithmic sigmoid (Eq. 3), which are among the most popular transfer functions, have been incorporated in the hidden and output layers, respectively 63,68 : Figure 2a,b show the general shapes of the hyperbolic tangent sigmoid and logarithmic sigmoid transfer functions, respectively. This figure indicates that the earlier provides a value between − 1 and + 1, while the latter produces a value ranging from 0 to + 1.
To this end, it is necessary to normalize both the independent (IV) and dependent variables (DV) into the [0 1] range using Eqs. (4) and (5), respectively.
NoD designates the number of datasets. X 1 , X 2 , and X 3 indicate the normalized value of the additive type, nanoparticle dose, and pressure. Moreover, Y stands for the normalized CO 2 permeability.

Evaluation of the model's accuracy
It is often mandatory to measure the deviation between experimental and predicted values of the dependent variable using statistical criteria. This study applies correlation coefficient (R), coefficient of determination (R 2 ), summation of absolute error (SAE), mean absolute error (MAE), absolute average relative deviation (AARD), and mean squared error (MSE). Accordingly, Eqs. (6) to (11) present the formula of R, R 2 , SAE, MAE, AARD, and MSE, correspondingly 71 .
The above equations need experimental ( DV exp ) and calculated ( DV cal ) dependent variables as well as the average value of the DV exp . Equation (12) calculates this average value, i.e., DV exp .

Results and discussions
This section introduces the results of relevancy analysis by MLR, MLP-ANN development, and statistical and graphical investigations of the proposed model.

Relevancy analysis by the multiple linear regression.
Before constructing the MLP-ANN to estimate the CO 2 permeability in PMP/nanoparticle membranes, the relevancy between dependent and dependent variables must be explored. The MLR is a well-known method in this field 72 . Equation (13) is a simple MLR model that correlates the normalized CO 2 permeability ( Y cal ) to the normalized values of the independent variables based on 112 experimental datasets.
The positive sign of the X 1 , X 2 , and X 3 coefficients suggests the direct dependency of CO 2 permeability on the involved independent variables. Also, the coefficient magnitude shows the strength of the relationship between the dependent and independent variables. As Fig. 3 illustrates the CO 2 permeability in PMP/nanoparticle membranes has the strongest dependency on the nanoparticle dose and the weakest dependency on the additive type.
The observed AARD = 88.24%, R 2 = 0.40145, and SAE = 7634.84 barrer between experimental CO 2 permeabilities and MLR predictions show that the considered problem is mainly governed by a nonlinear model.
The accuracy of indices is calculated after de-normalizing the MLR prediction for the normalized CO 2 permeability using Eq. (14).
Nonlinear modeling by the MLP-ANN. The general topology of the MLP-ANN to relate the CO 2 permeability in PMP/nanoparticle MMMs has been shown in Fig. 4.
This stage constructs 90 MLP-ANN approaches with different numbers of hidden neurons. Indeed, these MLP-ANN models may have one to nine neurons in their hidden layers. In addition, the MLP-ANN with a specific number of hidden neurons is trained and tested 10 different times. www.nature.com/scientificreports/ Figure 5 shows the results of ranking the 90 constructed MLP-ANN models. Generally, the MLP-ANN accuracy increases (rank decreases) by increasing the number of hidden neurons. This observation is related to the increasing MLP-ANN size as well as the number of their weights and biases. The figure indicates that the second-developed MLP-ANN with eight hidden neurons (rank = 1) is the best model for estimating the CO 2 permeability in PMP/nanoparticle MMMs. In addition, the 9th-built MLP-ANN with only one hidden layer is the lowest accurate model (rank = 90) for the considered task.
The best MLP-ANN is applied to accomplish all subsequent analyses and the remaining 89 models are ignored. Figure 6 presents the general shape of the MLP-ANN approach constructed to estimate the CO 2 permeability in MMMs. It can be seen that the MLP-ANN has only one hidden layer with eight neurons, i.e., 3-8-1 topology. The hyperbolic tangent sigmoid and logarithmic sigmoid transfer functions can also be seen in the hidden  www.nature.com/scientificreports/ and output layers. It should be noted that the modeling phase of the CO 2 permeability in both PMP and PMP/ nanoparticle membranes is done in the MATLAB environment (Version: 2019a) 73 . Table 3 reports the achieved accuracy of the proposed MLP-ANN in the training and testing stages. This table also shows the accuracy of the built MLP-ANN model for predicting the CO 2 permeability of the overall datasets. Five statistical criteria (i.e., R, MAE, AARD, MSE, and SAE) have been used in this regard. All these accuracies are acceptable enough from the modeling point of view.
Performance checking. The cross-plot which graphically inspects the linear correlation between experimental and predicted values of a dependent variable is a practical method to evaluate the reliability of datadriven models. Figure 7a-c illustrate the linear correlation between experimental CO 2 permeabilities and their associated calculated values by the MLP-ANN approach. Since both training and testing datasets are mainly located around the diagonal lines, the MLP-ANN reliability is approved by the visual inspection. Moreover, the closeness of the correlation coefficients of the training, testing, and all datasets to R ~ 1 (i.e., 0.99658, 0.98433, and 0.99477) is another indication of the MLP-ANN model.
The actual and predicted CO 2 permeabilities in the pure PMP membranes and PMP/nanoparticles MMMs in the training, as well as testing stages are depicted in Fig. 8. This analysis justifies the outstanding performance of     www.nature.com/scientificreports/ also accurately learns the increasing effect of the filler dose on CO 2 separation by the membrane-based process.
Increasing the CO 2 permeability in membranes by increasing the filler dose was also previously forecasted by the MLR relevancy investigation. The literature has related this permeability improvement to the alumina-polymer interactions and pore volume increment due to the Al 2 O 3 presence within the polymer chain 61 .
The effect of working pressure on CO 2 separation by the PMP/ZnO membranes with five nanoparticle concentration levels (2.5, 5, 8, 10, and 15 wt%) has been presented in Fig. 10. This figure displays both laboratorymeasured CO 2 permeabilities and their related MLP-ANN predictions. An excellent agreement between the experimental and modeling permeability-pressure profiles is easily observable through this investigation. The MLP-ANN also correctly identifies the pressure as well as the filler effect on CO 2 permeability in PMP/ZnO mixed matrix membranes.
As expected, the CO 2 permeability in the mixed matrix membranes rises by increasing the working pressure. This observation is in a direct relationship with the driving force improvement due to the pressure enhancement.
The effect of filler type (ZnO, Al 2 O 3 , TiO 2 , and TiO 2 -NT) on the CO 2 separation ability of PMP-based membranes in the same working pressure is illustrated in Fig. 11. It can be seen that different fillers represent various roles in CO 2 -MMM interaction. Indeed, the PMP/TiO 2 and PMP-TiO 2 -NT provide the CO 2 molecule with minimum and maximum permeabilities within the membrane structure. The literature justified the higher CO 2 permeability in PMP-TiO 2 -NT to the free volume expansion and porosity increase due to the functionalized nanoparticle presence in the membrane body 60 .

Conclusions
This study uses a two-step methodology, i.e., multiple linear regression and multilayer perceptron artificial neural networks to simulate carbon dioxide permeability in mixed matrix membranes. The carbon dioxide permeability in pure poly(4-methyl-1-pentene) and PMP/nanoparticle membranes (i.e., PMP/ZnO, PMP/Al 2 O 3 , PMP/ TiO 2 , and PMP/TiO 2 -NT) has been studied based on 112 experimental datasets collected from the literature. The multiple linear regression method applies to anticipate the dependency of the carbon dioxide permeability on the membrane composition (additive type and dose) and pressure. This method shows that the carbon dioxide permeability is directly related to all independent variables and it has the strongest correlation with the nanoparticle dose in membrane structure. The MLP-ANN is then utilized to construct a non-linear approach to estimate the carbon dioxide permeability as a function of additive type, nanoparticle dose, and pressure. This MLP-ANN with the 3-8-1 topology predicted 112 experimental carbon dioxide permeabilities in the involved MMMs with excellent accuracy (i.e., R = 0.99477, MAE = 6.87, AARD = 5.46%, MSE = 152.75, and SAE = 769.68).
The modeling results clarify that the PMP/TiO 2 -NT has a better carbon dioxide separation than the PMP/ZnO, PMP/Al 2 O 3 , and PMP/TiO 2 mixed matrix membranes. Finally, the obtained results in this work demonstrated the excellent potential of the ANN for estimating the separation factors of mixed matrix membranes for carbon capture and sequestration applications.

Data availability
All the literature datasets analyzed in this study are available at a reasonable request from the corresponding author (S.A. Abdollahi).  Figure 11. The effect of additive type on the CO 2 permeability in PMP/nanoparticle membranes.