Development of ANFIS technique for estimation of CO₂ solubility in amino acids and study on impact of input parameters

1 Introduction

Removal and capture of pollutants from water and air has been investigated recently to meet the global standard target as set out by international organizations. Among different pollutants, carbon dioxide has been on the main focus for controlling the climate change (Cheng, 2021; Liu, 2022; Shan, 2022; Batres, 2021; Schweizer, 2020). Chemical sorption of CO₂ in solvents has been the primary process for CO₂ capture in chemical industries such as oil and gas industry. In this process, CO₂ molecules are transferred from the gas phase to the liquid phase by mass transfer and chemical absorption takes place in the liquid phase for removal of CO₂ from gas phase. Indeed, the presence of chemical reaction enhances the removal efficiency drastically in this process (Dhaneesh and Ranganathan, 2022; Fang, 2020; Li, 2022; Pang, 2021). For design of the CO₂ capture process based on chemical absorption, other criteria should be taken into account such as the process economic and also regeneration of the solvent for reuse in the process. Furthermore, the loading of CO₂ in the solvent is the key parameter which should be observed for design of process and also selecting the proper solvent for the separation process.

CO₂ solubility in wide solvents can be determined through experimental measurements which is tedious and costly to be done. On the other hands, computational techniques can be used to estimate the CO₂ solubility in solvents as function of process parameters and the absorbent properties, as well. In the modeling techniques, models based on machine learning can be employed to predict the solubility of CO₂ in wide solvents and wide range of parameters (Soroush, 2019). The method of machine learning has attracted much attention in the area of CO₂ capture in chemical solvents due to its superior accuracy and better fitting compared to similar models (Jin, 2022; Shalaby, 2021; Wang, 2021; Yin, 2022; Zhang, 2022). However, these methods rely on the available data for training and some measurements are required to provide the CO₂ solubility data to build the machine learning models. This can be regarded as the disadvantage of these models, as they are not recognized as pure predictive models for CO₂ removal using chemical solvents.

Because of a recent development in technology known as machine learning, computers are now capable of learning from data without the need for explicit programming. This is one of the many applications of computer science (ML). One of the primary objectives of machine learning is to develop meta-programs that are able to analyze experimental data and then make use of that data as an input in the construction of models that make use of the same experimental data as an input (El Naqa and Murphy, 2015; Goodfellow et al., 2016; Shang, 2021; Tian, 2022; Alibak, 2022).

An optimal neural network approach to solving the function approximation problem based on neural networks is adaptive network-based fuzzy inference systems (ANFIS). This type of information system provides a mapping relationship between the input data and the output data through the use of crossbred learning methods. This is in order to identify the optimum distribution of membership functions for the input data. In the ANFIS architecture, artificial neural networks (ANNs) are employed in conjunction with fuzzy logic (FLs). The ANFIS modeling process can become more systematic and less dependent on expert knowledge when such a framework is used. In order to construct this system of inference, five layers are used. There are several nodes in every ANFIS layer, and each of these nodes is defined by a node function.

In this research, we have used the ANFIS method, but this method has already been used for the same data set in (Soroush, 2019). The difference of our work is that we have used metaheuristic methods including DE and FA to optimize this model. Therefore, the optimized ANFIS model is different from its raw model and its result is more reliable in terms of over-fitting and generality. The data for the modeling have been collected on the solubility of CO₂ in amino acids salt solutions.

2 Material and methods

2.1

2.1 Data set of CO₂ solubility

The dataset in this research have more than 800 data points that is taken from (Soroush, 2019) for loading CO₂ into amino acid salt solutions. In the dataset, there are six input features including T(K), weight (%), PCO₂(kPa), Mw_am, Mp C, and Mw_C and the single output is alfa which is the values of CO₂ loading. The explanation of the parameters can be found in (Soroush, 2019) in which the data of inputs are related to the operational parameters of the absorption (e.g., P and T) as well as the physio-chemical parameters of the absorbent such as molecular weight and percentage of salt. In fact, the most important parameters have been taken into the model for prediction and description of the CO₂ loading. The distribution of variables is shown in Fig. 1. In this figure, the diagonal subplots (with the x-axis and y-axis identical) also show kernel density estimates (KDEs). Similar to histograms, KDE plots show the distribution of observations.

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Fig. 1

Distributions of variables used in this work.

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2.2

2.2 Adaptive network-based fuzzy inference system (ANFIS)

The base model employed in this work for correlation of the CO₂ solubility data is ANFIS model. The ANFIS network is famous for modeling complex systems using a hybrid neuro-fuzzy approach (Jang and Sun, 1995; Jang, 1993). With ANFIS, a collection of fuzzy If-Then conditions (rules) is used to incorporate a fuzzy system's reasoning style, which is similar to human reasoning. As universal estimators, ANFIS models offer interpretable If-Then rules as well as universal approximation (Azeem, 2012; Sadrmomtazi et al., 2013).

We consider in this system only two inputs × and y alongside with a single output f_out to illustrate the procedures of an ANFIS. Fig. 2 shows the ANFIS architecture that was used in this work. Below is a description of the node functions associated with each layer (Ziari, 2016).

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Fig. 2

Architecture schematics for ANFIS used in correlation of CO₂ loading data.

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Layer 1: In this layer, the nodes are adaptive and can do tasks including as (Azeem, 2012): $$O_{1\text{,}i}={\mu }_{A_i}\left(x\right)\text{for}i=1\text{,}i=2$$ $$O_{1\text{,}i}={\mu }_{B_{i-2}}\left(y\right)\text{for}i=3\text{,}i=4$$

In this case, x (or y) represents the node's input. The linguistic label is A_i (or B_j), μ(x)/μ(y) represents the membership function. Most often, a bell-shaped distribution is used with an upper bound of 1 and a lower bound of 0.

Layer 2: The nodes in second layer are fixed nodes, outlined in circles and labeled Π, whose functions will be multiplied by input signals to generate an output signal (Azeem, 2012): $$O_{2,i}={\mu }_{A_i}(x)·{\mu }_{A_i}(x)=w_{i}fori=1,2$$

A rule's firing strength is represented by the output signal w_i.

Layer 3: This layer relies on fixed nodes, represented by circles and labeled N, as well as a node function to Calculate the ratio of each node's firing strength to the sum of all rules' firing strengths to normalize firing strength (Azeem, 2012): $$O_{3\text{,}i}=\frac{w_i}{\sum w_i}=\frac{w_i}{w_1+w_2}\text{for}i=1\text{,}2$$

Layer 4: Adaptive nodes are indicated by a square as a part of this layer, and node functions are defined for each of these nodes (Azeem, 2012): $$O_{3,i}={\overline{w}}_i·f_{i}fori=1,2$$

In this case, f₁, f₂ are the if-then fuzzy rules as (Azeem, 2012):

• Rule 1: if (x==A₁ & y==B₁):

$$f_1=p_{1}x+q_{1}y+r_1$$

• Rule 2: if (x==A₂ & y==B₂):

$$f_2=p_{2}x+q_{2}y+r_2$$

Here, the parameters set, or consequent parameters, are [p_i, q_i, r_i] (Chen et al., 2010).

Layer 5: There are four fixed nodes in this layer, each indicated by one circle and tagged with the Sigma, with a node function to calculate the total output (Azeem, 2012): $$O_5=\sum {\overline{w}}_i·f_i=f_{\mathit{out}}fori=1,2$$

In ANFIS, the error signal is calculated recursively from the output layer to the input layer using backpropagation gradient descent. The backpropagation learning rule is the same for feedforward neural networks (Topçu and Sarıdemir, 2008; Ramezanianpour, 2004).

3 Results and discussions

The model optimization is the main novelty of this research as mentioned above. The result of this step is shown in Fig. 3. This figure shows that the FA optimization makes the best performance on near to 180 iterations. So, the final ANFIS model is selected at the hyper-parameters of this setting. The future analysis is done based on this model.

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Fig. 3

Tuning with metaheuristics.

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Fig. 4 shows the comparison of expected and predicted values by the model developed and tuned in this study. The red points are the test data, and the blue points are the training data, which are randomly separated with a ratio of 2 to 8. The distance of each point from the diagonal line indicates its prediction error. By interpreting this figure, we realize that the obtained model has high generality and accuracy because the test points are not too far apart from the training points. Indeed, the model has been well tuned such that the CO₂ loading data can be well estimated using the developed model.

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Fig. 4

Actual and Estimated Values with ANFIS Model.

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The same fact is confirmed by Table 1 because the error values as well as the R² score in testing and training are close to each other. Indeed, the statistical analysis is reported in Table 1 indicating that the R² values for both training and validation of the model are greater than 0.95 which confirms the goodness of fitting data points.

Fig. 5 compares the relative importance values of features on target output. Permutation is used to determine the importance of features in this case. If the data are tabular, any fitted estimator can be examined using permutation feature importance. For non-linear estimation methods or opaque methods, it will be beneficial. Permutation feature importance describes the effect of randomly shuffling a single feature value on the model score. Using this method, there is no connection between the feature and the destination. Therefore, if the model's score goes down, it means it is overly dependent on that attribute. It can be also seen that the partial pressure of CO₂ in the gas feed plays important role in the amount of CO₂ loading in the solvent as predicted by the model. This is in agreement with the physical observation as the mass transfer of CO₂ from gas phase to the absorbent phase is controlled and driven by the partial pressure gradient between two phases.

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Fig. 5

Permutation importance of features.

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4 Conclusion

As part of this investigation, we devised a sophisticated model for estimating the amount of carbon dioxide that can be loaded onto amino acid salt solutions. This model employs a metaheuristic optimized ANFIS that is derived from a diverse selection of amino acids. The innovative aspect of this research is the utilization of the Differential Evolution (DE) and Firefly Algorithm (FA) metaheuristics in order to address the hyper-parameter tuning of ANFIS as an optimization problem that is based on differential evolution. As a consequence of this, the ANFIS model's R2 score for the test data is 0.9520, whereas its score for the training data is 0.9841. This demonstrates that the model that was proposed is both comprehensive and reliable in terms of the predictions it makes. The closeness of scores and errors in testing and training shows the lack of over-fitting of the models, which has not been investigated and analyzed in many similar studies. The RMSE error rate is 1.17E-01 whereas the MAPE error rate is 1.14E-01. Both of these error rates are almost identical to each other.