The multiverse optimizer (MVO) is a new method of an intelligent optimization algorithm proposed by Mirjalili et al. (2016), which has been successfully applied in various function optimizations and engineering designs (Vivek et al., 2018; Hassan and Zellagui, 2019; Jain et al., 2019; Dao et al., 2020; Zhou et al., 2022). The multiverse theory in physics is the inspiration for the MVO algorithm. In the multiverse theory, the emergence of the individual universe is the result of a single giant explosion, and multiple giant explosions have contributed to the birth of the entire multiverse population. White holes, black holes, and wormholes are the three core concepts in the multiverse theory: white holes have strong repulsion and can release all objects; black holes have extremely high gravitational forces and can absorb all objects; and wormholes connect different universes and the orbit of the transported object.

MVO is based on the principle that matter in the universe transfers from white holes to black holes through wormholes. Under the combined action of white holes, black holes, and wormholes, the entire multiverse population will eventually reach a state of convergence. MVO classifies the search process into two stages: exploration and development. The exploration of the search space is completed through the exchange of black/white holes, and the development process is completed in the way of wormholes. MVO cyclically iterates the initial universe through white hole/black hole tunnels and wormholes, in which the universe represents the feasible solution to the problem, the objects in the universe represent the components of the solution, and the expansion rate of the universe represents the fitness value of the solution. The mathematical model of the algorithm is given as follows:

Multiverse initialization: A set of random universes U is created, as shown in Equation 1. U = x 1 1 x 1 2 ⋯ x 1 d x 2 1 x 2 2 ⋯ x 2 d ⋮ ⋮ ⋮ ⋮ x n 1 x n 2 ⋯ x n d . (1)

Here, d is the number of variables and n is the number of universes (candidate solutions).

Black and white hole mechanisms: Due to the different expansion rates of each individual universe, objects in an individual universe are transferred through white hole/black hole orbits. This process follows the roulette mechanism, as shown in Eq. 2. x i j = x k j r 1 < N I U i , x i j r 1 ≥ N I U i , (2)

where x i j is the i variable of the j universe. N I U i is the normalized expansion rate of the universe i . r 1 is a random number between 0 and 1, and x k j is the i variable of the j universe selected according to the spiral mechanism.

Wormhole mechanism: Without considering the size of the expansion rate, in order to achieve local changes and improve its own expansion rate, the individual universe will stimulate the internal objects to move to the current optimal universe as shown in Eqs 3–5. W E P = W E P min + l × W E P max − W E P max L , (3) T D R = 1 − l 1 / p L 1 / p . (4)

Here, W E P represents the probability of the existence of wormholes in the multiverse space, W E P min represents the minimum probability (0.2 in this study), W E P max represents the maximum probability (1 in this study), l represents the current number of iterations, and L represents the maximum number of iterations. T D R represents the travel distance numerical value, where p (equal to 6 in this article) defines the development accuracy in iterations. The higher the value of p , the faster the local development but smaller the scope of the search. x i j = X j + T D R × u b j − l b j × r 4 + l b j , r 3 < 0.5 , X j − T D R × u b j − l b j × r 4 + l b j , r 3 ≥ 0.5 , r 2 < W E P , x i j , r 2 ≥ W E P . (5)

Here, x i j represents the j object of the current optimal universe and u b j and l b j refer to the lower and upper bounds of x i j , respectively. r 2 , r 3 , and r 4 are random numbers in the range [0, 1].

In the MVO, the update of the individual mainly depends on the size of the expansion rate and is then randomly updated according to the current global optimal universe and wormhole existence probability (WEP) parameters. Since the optimal value in the early stage of the algorithm is often too far from the true value, using the global optimal universe and the updated strategy will increase the probability of the algorithm falling into the local optimal and may cause the algorithm to slow down the convergence speed. Therefore, this study introduces opposition-based learning (OBL) to improve the global search ability and algorithm stability of the MVO.

OBL was proposed by Tizhoosh (2005) to improve the convergence stability and global search ability of other algorithms by considering opposite anti-population, opposite weight, anti-behavior, anti-exploration, anti-exploitation, etc. Currently, some scholars have introduced OBL into optimization algorithms such as PSO (Wang et al., 2011), WOA (Ewees et al., 2018), SCA (Gupta and Deep, 2019), and SSA (Tubishat et al., 2020). In this study, OBL is introduced into MVO, and the OBL-MVO algorithm is established for optimizing ensemble learning. The difference between the OBL-MVO and MVO lies in three stages: negative universe initialization, anti-black hole mechanism and anti-white hole mechanism, and anti-wormhole mechanism. The OBL-MVO algorithm includes a total of six stages: initialization of the positive universe and negative universe, calculation of the expansion rate (fitness value) of the universe, black hole and white hole mechanism, anti-black hole mechanism and anti-white hole mechanism, wormhole mechanism, and anti-wormhole mechanism. The positive universe and negative universe initialization run only once, and the remaining five stages are executed in a loop. The conceptual model of the proposed algorithm is given in Figure 1.

The following section introduces the initialization of the positive universe and the negative universe, the anti-black hole mechanism, the anti-white hole mechanism, and the anti-wormhole mechanism.

Anti-multiverse establishment: In order to solve the situation of the optimal value of the MVO algorithm often being too far from the real value in the early stage, the global search ability and convergence speed of the algorithm are improved. From the establishment stage of the multiverse population, the universe is divided into a positive universe and negative universe (the number of populations is consistent with the original MVO algorithm), and the positive universe is established by Eqs 6–9: U = x 1 1 x 1 2 ⋯ x 1 d x 2 1 x 2 2 ⋯ x 2 d ⋮ ⋮ ⋮ ⋮ x n / 2 1 x n / 2 2 ⋯ x n / 2 d , (6) x ′ i j = u b j − x i j + u b j , (7) U ′ = x ′ 1 1 x ′ 1 2 ⋯ x ′ 1 d x ′ 2 1 x ′ 2 2 ⋯ x ′ 2 d ⋮ ⋮ ⋮ ⋮ x ′ n / 2 1 x ′ n / 2 2 ⋯ x ′ n / 2 d , (8) U a l l = U U ′ . (9)

Here, x i ′ is the corresponding negative universe in the positive universe x i and x ′ i j is the anti-object of the corresponding object in the negative universe.

Anti-black hole mechanism and anti-white hole mechanism: The material exchange mechanism between black holes and white holes is the exploration of the search space by MVO, but there are problems of the local search accuracy being too high and global search ability being insufficient. Therefore, the global search ability and overall stability of the algorithm are increased by setting the reverse black hole and reverse white hole mechanisms. The reverse black hole and reverse white hole mechanisms include random black hole and white hole stage and reverse object propagation stage. Both stages are controlled by a convergence factor λ that decreases linearly with the number of iterations to control the probability of occurrence. The random black hole and white hole stages are shown in Eq. 11, and the reverse object propagation stage is shown in Eq. 12: λ = 1 − l L , (10) x i j = x k j , r 7 < λ , x i j , r 7 ≥ λ , (11) x i j = x k j , r 5 < 0.5 , x i j , r 5 ≤ 0.5 , r 6 < λ , x i j , r 6 ≥ λ , (12) where λ is a convergence factor that decreases linearly with the number of iterations. r 5 and r 6 are random numbers in the range [0, 1].

The anti-wormhole mechanism: The anti-wormhole mechanism differs from the wormhole mechanism, where the anti-wormhole mechanism improves the global search by searching backward from the universe with the lowest expansion rate. To prevent this mechanism from affecting convergence, internal objects are, therefore, only stimulated to move toward the current worst universe if it is determined that the anti-wormhole mechanism is favorable to the increase in the universe’s expansion rate. x i j l + 1 o p p o s i t e = x i j + X j ′ × T D R × u b j − l b j × r 4 + l b j , r 3 < 0.5 , x i j + X j ′ × T D R × u b j − l b j × r 4 + l b j , r 3 ≥ 0.5 , r 2 < W E P & r 6 < λ , x i j , r 2 ≥ W E P ∥ r 6 < λ , (13) x i j l + 1 = x i j l + 1 o p p o s i t e , N I x i j l + 1 o p p o s i t e > N I x i j l , x i j l , N I x i j l + 1 o p p o s i t e ≤ N I x i j l , (14) where X j ′ denotes the j object of the current worst universe. x i j l + 1 o p p o s i t e denotes the result of the calculation based on the reverse wormhole mechanism, x i j l + 1 is the j variable of the i universe where the number of iterations is l + 1 , and x i j l is the j variable of the i universe before the reverse wormhole mechanism in the l iteration.

The idea of ensemble learning is that even if one base learner makes a wrong prediction, other base learners can correct the error, which aims to integrate multiple base learners to improve the accuracy of prediction. The main process is to first train multiple base learners by certain rules, then combine them using an integration strategy, and finally predict the result by comprehensive judgment of all base learners. Currently, ensemble learning has been successfully applied in pattern recognition, text classification, numerical prediction (Wang et al., 2011), and other fields.

The current integration strategy can be broadly divided into two categories: one is the sequential generation of base classifiers, with strong dependencies between individual learners, represented by AdaBoost (Ying et al., 2013), and the other is the parallelized integration method, which can generate base classifiers simultaneously, without strong dependencies between individuals, represented by Bagging (Dudoit and Fridlyand, 2003).

As unconventional reservoir data are collected from different reservoirs and have different reservoir data characteristics, this study improves the accuracy of the model by accommodating as many base learners as possible into a parallelized integration method to suit the different reservoir data characteristics. However, accommodating more models can lead to overfitting (Džeroski and Ženko, 2004). Therefore, the K-fold cross-validation method and opposition-based learning are introduced to avoid overfitting.

In this study, the search for the best weights of base learners is considered an optimization problem, and the OBL-MVO algorithm is used to solve the optimization problem. The integration strategy of AIL-OMO is to divide the original dataset into several sub-datasets which are fed into each base learner in layer 1. In layer 1, each base learner outputs its own prediction results as meta features. The meta features are then used as an input to layer 2, where the OBL-MVO is used to search for the best weights of base learners.

The integration strategy used in this study is shown in Figure 2.

In order to fully exploit the individual strengths of each model, the probabilities of each reservoir predicted by each model are used as an input to train the ensemble learning model. In order to improve the accuracy and generalize the ability of the models completed by integration learning, the aforementioned OBL-MVO algorithm is introduced to reasonably optimize the integration learning process and search for the best weight of base learners.

The main contributions of this study are as follows:

(1) Collection of data and work on data collation and labeling.

In this section, the data used in this paper and the preprocessing of the data are introduced. In order to build a reservoir classification model that can adapt to various unconventional features, 12,379 pieces of reservoir fluid property data from eight different wells were used for training the model. The data contain six categories of labels: dry layer, poor oil layer, water layer, oil-bearing water layer, oil–water layer, and oil layer. The data include AC (sonic interval transit time), CNL (compensated neutron logging), DEN (compensated density), GR (natural gamma ray), SP (spontaneous potential), CAL (caliper logging), and R045 (0.45 m potential resistivity). Figure 4 shows the degree of correlation between variables in reservoir fluid property data (excluding the oil field name). After obtaining the collected data, the annotation work of the dataset was carried out by analyzing the logging curves as the training data for the model.

In the data preprocessing stage, this study mainly operates on the following issues:

(1) Rejection operations for data that are missing or do not conform to common sense.

In this section, OBL-MVO is used to solve the optimization problem to build adaptive ensemble learning classifiers for reservoir fluid property prediction. Since the dataset of unconventional reservoirs contains various unconventional reservoir features, each base learner is differently adapted to the unconventional reservoir features. Therefore, the weight searching problem in the integration strategy is abstracted as an optimization problem to obtain and integrate a model with generalization performance on all unconventional reservoir features in the dataset. The OBL-MVO algorithm is used to optimize the weights of the base learners based on the K-fold cross-validation test set to build the AIL-OMO to avoid the overfitting of partial features for unconventional reservoirs. In this optimization problem, to avoid overfitting of the local features of unconventional reservoirs, the accuracy of the validation set in K-fold cross-validation of each base learner for each unconventional reservoir feature is obtained from the validation set in K-fold cross-validation. In this case, the weights of the base learners are encoded into the individual dimensional attributes of the population, as shown in Eq. 15. The degree of learning of each model in each unconventional reservoir feature is involved in the calculation of the adaptation value of the optimization algorithm, as shown in Eq. 16. X i = x i 1 x i 2 ⋯ x i d , x i j ϵ 0,1 1 ≤ j ≤ d , (15) f i t n e s s = 1 − ∑ i = 1 t s max ∑ j = 1 dim x i j ∙ x i j − y t u r e i t s , (16)

where X i represents the individual i ; d is the dimension vector of X i ; x i j is the j t h dimension value of the individual i , representing the weight of the j t h model; y i j is the classification possibility of the j t h model in the test set of K-fold cross-validation; y t u r e i is the true classification of the i t h sample; t s is the number of samples in the full test set in K-fold cross-validation; and f i t n e s s is the fitness function set by this study.

In order to show the performance of OBL-MVO on this dataset, OBL-MVO is compared with the other swarm intelligence heuristic algorithms, including whale optimization algorithm (WOA), gray wolf optimization (GWO), and MVO.

Among them, the parameter settings of each swarm intelligence heuristic algorithm are shown in Table 1.

In order to compare the four intelligent optimization algorithms, it is necessary to compare the fitness value curves of their optimization results. Since the population size of different algorithms affects the convergence of the fitness curve, in order to balance the effect of the population size of each algorithms on the fitness value, the population size for each algorithm is set to 20, 50, 100, and 200. The fitness curve results of GWO, WOA, MVO, and OBL-MVO are shown in Figure 5.

Figure 5 shows that OBL-MVO has a faster convergence rate than MVO, WOA, and GWO (the number of iterations required for the fitness function to converge and stabilize is the least); OBL-MVO shows a better convergence fitness value than MVO, WOA, and GWO, indicating that OBL-MVO has a better optimization effect than MVO, WOA, and GWO on the dataset. The corresponding weights of base learners optimized by OBL-MVO adaptive optimization are shown in Table 2.

In this section, the predicted results of the AIL-OMO model and evaluation of its performance are presented. The AIL-OMO model was trained using the training dataset and evaluated on the test set. This paper adopts the accuracy rate as the evaluation metric, which is widely used in reservoir classification, lithology identification, and reservoir fluid identification (Moffatt and Williams, 1998; Al-Anazi and Gates, 2010; Boyd et al., 2013; Onwuchekwa, 2018).

Figure 6 depicts the results of using seven machine learning algorithms to predict the reservoir fluid, including the AIL-OMO model proposed in this paper. The accuracy of the AIL-OMO model surpasses that of the other six models, achieving an accuracy rate of 92.75%, followed by the SVC and linear SVC models in well 1. Figure 7 presents the reservoir fluid prediction results of the seven machine learning models in well 2. The accuracy rates from high to low are given as follows: AIL-OMO (81.93%), logistic regression (70.37%), linear SVC (69.30%), linear discriminant analysis (69.26%), KNN (69.11%), AdaBoost (66.45%), and SVC (64.37%). Figure 8 shows the reservoir fluid prediction results of the seven machine learning models in well 3. Among the prediction results of each model, the accuracy rate of the AIL-OMO model is better than that of the other models, with an accuracy rate of 88.72%, followed by SVC with an accuracy rate of 83.03%. The worst model is AdaBoost, with an accuracy rate of only 65.57%. From the experimental results, it can be interpreted that integrating multiple weak learners into one strong learner using the OBL-MVO algorithm leads to a significant improvement in the prediction accuracy of the original model. Therefore, the proposed AIL-OMO model has high accuracy and is more conducive to the prediction of reservoir fluid.

Figure 9 shows the change in the classification accuracy of the AIL-OMO model with different party sizes.

Figure 9 shows the classification accuracy of different reservoir fluids by AIL-OMO. Each value in the figure represents the ratio of the number of reservoir fluids identified by AIL-OMO to the actual number of reservoir fluids in the dataset. Figure 6 shows that the results of AIL-OMO exhibit slight fluctuations under different party sizes. The classification accuracy for the poor oil layer, water layer, and oil-bearing water layer is relatively low. This can be attributed to the smaller percentage of poor reservoirs, water layers, and oil-bearing water layers in the dataset.

Table 3 presents the comparison results of the accuracy of AIL-OMO and the other six machine learning models. Among them, AIL-OMO with a party size of 100 achieves the highest accuracy of 85%. This model demonstrates the fewest classification errors for each reservoir. However, when the party size is increased to 200, the accuracy of AIL-OMO decreases to 78%. This decrease in accuracy can be attributed to the party size being too large, causing overfitting of the model to the data.