Parallel and distributed chimp-optimized LSTM for oil well-log reconstruction in China

Abstract

Well-log analysis contributes significantly to effective oil and gas extraction, but inconsistent logs may render subsequent geological analyses useless. This study tackles this problem by devising a deep Long Short-Term Memory (LSTM) model that uses the new Parallel and Distributed Chimp Optimization Algorithm (PDCOA). PDCOA’s primary goal is to speed up the process of hyperparameter tuning for LSTMs by letting them work in parallel and across multiple computers, with separate groups of computers communicating with each other regularly to ensure the system is diverse and reliable. It is designed for reconstructing missing well-log data, showing that the proposed method is more scalable, efficient, and accurate as a predictor. This feature makes it a valuable tool for geological interpretation and estimating hydrocarbon resources.

Introduction

The process of oil and natural gas extraction from boreholes is quite intricate and requires knowledge of the underlying structures^1,2,3. As a formidable source of geological information, well-log data enables the assessment of the likelihood and the amount of hydrocarbon resources that are located under the Earth’s crust^4,5,6. These logs are essential for the proper construction of reservoir models⁷, production forecasts⁸, and economic evaluations^9,10,11. Nevertheless, the quality or completeness of well logs can be a serious problem^12,13. Unavailability or incompleteness of such data can significantly impair geological interpretation, making it a challenge to predict resources and plan drilling operations^14,15.

Recently, there has been a focus on machine learning^16,17,18 and deep learning approaches¹⁹, especially LSTM networks, due to their performance in sequential data modeling and prediction in different domains²⁰. LSTMs are a class of recurrent neural networks that are very effective in applications involving time- or sequence-boosted data, such as well-log data. Furthermore, their innate characteristics allow them to also deal with long-term dependencies and intricate, massive structures in sequences, making those networks suitable for predicting missing or deficient sequences. This particular characteristic is crucial in well-log analysis since the databases are usually incomplete due to, for instance, poor measurements and broken sensors²¹. The intelligence of LSTM networks in dealing with long-range connections means that they can complete unobserved data points, which contributes to better comprehensive geological interpretations and enhances overall data quality²².

Nevertheless, the performance of LSTM networks depends to a high degree on the selection of hyperparameters, hidden units, learning rate, and dropout rate, for instance. A performance can be optimal if these parameters are tuned appropriately, but tuning them can be lengthy and resource-intensive. This situation is where algorithms for optimization become very useful²³. The Chimp Optimization Algorithm (COA)²⁴, which is based on the social behavior of chimpanzees, is a cost-effective, innovative optimization strategy that imitates the interactions between leader, follower, and chaser chimps^{25,26,27,28,29,30}. COA has effectively and successfully explored large, complex search spaces to improve on different optimal solutions for various optimization problems³¹.

Considering that LSTM network hyperparameter optimization is tedious, we took another turn integrating LSTM networks with the PDCOA. PDCOA was chosen because it is suitable for resolving real-world issues within reasonable time spans due to its parallelized and distributed computing systems. In essence, PDCOA splits the pools of potential answers into smaller groups and assigns them to different processors that bias evaluations and updates according to locality. Coordinating typical processors from different subpopulations ensures that high-quality solutions are combined to enhance the search for optimal LSTM configurations. This feature, therefore, reduces the requisite time for hyperparameter tuning and, at the same time, increases the robustness of the optimization, which is an ideal tool for complex LSTMs and large datasets.

Considering the potential that LSTM networks have in analyzing sequential data and evaluating the hyperparameter tuning capabilities of the PDCOA, this study seeks to enhance the performance of well-log predictions and the interpolation of missing values. Combining these two approaches offers possible abusive oil and gas exploration opportunities by providing more accurate and efficient resource estimation and geological evaluation.

The primary significance of the paper can be summarized as follows:

• Development of PDCOA: The primary novelty of this work is developing a parallel and distributed version of the Chimp Optimization Algorithm (PDCOA). Adding parallel and distributed computing techniques to the original Chimp Optimization Algorithm dramatically improves its efficiency and scalability, optimizing computational time and performance.

• Integration of PDCOA with LSTM Networks: Integrating PDCOA with LSTM networks facilitates more efficient hyperparameter optimization. Compared to traditional practices, this integration significantly reduces time and computational resources for hyperparameter tuning.

• Application to Well-Log Data Reconstruction: The model reconstructs missing well-log data during oil and gas exploration using LSTM and PDCOA. The model enhances the efficacy of geological evaluations due to LSTM’s long-term dependency modeling capabilities and PDCOA’s optimization capabilities.

• Parallel and Distributed Computing: Applying PDCOA to parallel and distributed computing increases the methodology’s scalability to larger datasets and real-time applications, achieving significantly enhanced efficiency compared to conventional optimization methods.

• Comparative Performance Evaluation: The LSTM-PDCOA model has outperformed numerous other LSTM-based optimization methods, such as LSTM-COA, LSTM-DLBMPA, LSTM-QCOA, and even other traditional machine learning models like XGBoost, LightGBM, and AdaBoost, in terms of prediction accuracy, missing data tolerance, and computation efficiency.

The rest of the paper is organized as follows: “Related works” represents the most related works. Section “Methodology” presents the methodology. Section “Experimentation” presents the experiment results. Section 5 discusses the results. Finally, “Discussion” concludes the paper.

Related works

Integrating machine learning (ML) techniques can dramatically improve the efficiency of drilling boreholes in oil and gas reservoir exploitation. These technologies optimize various processes associated with the drilling operation, ranging from estimating subsurface features to designing appropriate drilling parameters and managing risk based on operational constraints. Well-logs contain crucial sections that describe the characteristics of the subsurface, which often have gaps or missing values. Lawal et al.³² show that gradient boosting, among other models, can fill some of these gaps, increasing the commercial value of the well logs.

One of the most consistently mentioned applications of ML for borehole drilling is the rate of penetration (ROP) optimization³³. Borehole drilling is complicated by the need for consistent weight on the bit (WOB), rotary speed, and rock strength. In such situations, mechanistic modeling supports using random forests to link these parameters to Mean Squared Error (MSE), a metric quantifying the average squared difference between predicted and actual ROP values and ROP, thereby improving ROP by 31% and reducing MSE by 49%. Machine learning can also enhance real-time drilling decisions, optimize performance, and lower drilling operations’ costs.

ML models can avoid problems such as filtrate leaks, gas/oil/water shows, or sticking, resulting in wasted time in drilling. Neural network models and other ML algorithms can be applied to mitigate these problems; hence, more productive drilling time is achieved³⁴.

Advanced ML algorithms such as XGBoost, LightGBM, CatBoost, and AdaBoost can also be employed to estimate in situ stress from the cross-section of borehole breakout volumes. Such models have shown reasonably good precision in estimating the stress tensor, which is fundamental in the development of oil and gas fields³⁵.

ML algorithms can apply wellbore in a ‘real-time’ format, which can then be integrated into drilling assemblies³⁶. This method has been practically tested to drill complex 3-dimensional wellbores where the time is taken, and the accuracy of the drilling operations has improved³⁷.

ML processes can help to cut down on the number of structural models required to estimate the volume of Oil In Place (OIP)³⁸, thus facilitating the process. This improvement is done by defining the main parameters and retaining only the most representative models’ subsamples³⁹.

Machine learning improves the operational effectiveness of borehole drilling activities targeting oil and gas. These techniques generate economic, performance, and operational safety benefits in areas like well log data interpolation, drilling parameter optimization, operational issue forecasting and management, in situ stress reconstruction, autonomous drilling, and reduction of geological models. In summary, Table 1 compares previous studies utilizing LSTM enhanced by optimization algorithms with the proposed method.

Table 1 Comparison of previous studies utilizing LSTM enhanced by optimization algorithms.

Prior studies have looked at combining LSTM networks with different optimization algorithms, as listed in Table 1. Still, problems remained, such as high computation costs, lack of parallelism, and inadequate scaling to large datasets or noisy environments. For example, the standard COA⁴⁰ used for LSTM tuning demonstrated success in analyzing geophysical data but fell victim to sequential processing, which led to lengthy computation times. The same flaw can be found with the DLBMPA⁴¹, which improved the prediction of seismic data but did not have sufficient robustness against noise and a lack of scalability. On the other hand, our proposed PDCOA overcomes these problems by using parallel and distributed computing and partitioning the optimization tasks to be executed in parallel across several processors and subpopulations. This strategy reduces computational time while improving or maintaining the accuracy of the prediction. In addition, PDCOA’s coordination of subpopulations makes it more proficient at dealing with diverse noise. It makes it more robust, thus enabling it to outperform previous methods using complete or noisy well-log data. This research builds on earlier optimization strategies by improving the shortcomings of prior methods to reconstruct missing well-log data more effectively with robust scalability, efficiency, and accuracy.

Methodology

This section describes the methodology for reconstructing missing well-log data using LSTM networks optimized by a novel PDCOA. The approach integrates LSTMs’ sequential dependence modeling capabilities in well-log data and the hyperparameter tuning efficiency PDCOA offers through parallel and distributed computing. The process starts with PDCOA, which modifies the COA by splitting a population of potential solutions into subpopulations that can be processed in parallel across several computing nodes, thus speeding up optimization and improving scalability. These secondary solutions are configured with LSTM networks, which use temporal dependencies within the well-log dataset to predict missing data points. The following subsections provide deep mathematical models and implementation details for those readers who require further technical information.

Mathematical model for parallel and distributed COA (PDCOA)

Based on the social behaviors of chimpanzees, the COA is adept at intelligently searching intricate structures but is inefficient for large-scale problems. In this case, we propose the PDCOA, which divides the population into subpopulations processed simultaneously over several nodes, as illustrated in Fig. 1. This method improves computation time and scalability while periodic synchronization between nodes maintains adequate diversity and resilience in the solutions. The other sub-sections present the theoretical framework and policy guidelines.

Fig. 1

The whole block diagram of PDCOA.

Let us denote the population of chimps as X = {x₁, x₂, …, x_N} where xi = [x_i1, x_i2, …, x_iD] indicates the coordinates of the i-th chimp within a D-dimensional search space and, as usual, ‘N’ stands for number of individuals of the population. In the parallelized model, the population is divided into M subgroups (M ≤ N), as shown in Eq. (1)

$$X=\bigcup\limits_{{m=1}}^{M} {{X_m}}$$

(1)

where X_m stands for the group of subjects processed by the m-th processor.

Function f(x_i) is responsible for assessing the quality of each performed solution. According to Eq. (2), each processor evaluates fitness for its portion in the distributed arrangement.

$${f_m}({X_m})=\{ f({x_i})\left| {{x_i} \in {X_m}} \right.\}$$

(2)

The results obtained from multiple evaluations are combined to arrive at a single optimal result, as stated in Eq. (3).

$${x_{best}}=\arg \mathop {\hbox{min} }\limits_{{{x_i} \in X}} f({x_i})$$

(3)

The updating of the coordinates of every chimp is a function of the interaction with the leader (x_L), follower (x_F), and chaser (x_C) as specified in Eq. (4):

$$x_{i}^{{t+1}}=\alpha x_{L}^{t}+\beta x_{F}^{t}+\chi x_{C}^{t}+\delta x_{R}^{t}$$

(4)

In a parallel model, each processor is concerned with the position update of the individuals in its set X_m, and the updates are carried out according to Eq. (5), so α, β, x, and δ are influence coefficients. Also, x_R is used for exploration as it is a randomly generated position:

$$x_{{i,m}}^{{t+1}}=\alpha x_{L}^{t}+\beta x_{F}^{t}+\chi x_{C}^{t}+\delta x_{R}^{t}{\text{ }}\forall {x_i} \in {X_m}$$

(5)

In the case of distributed computing, the population is subdivided into non-interacting subpopulations (islands), which are evolving in parallelism. We occasionally swap the best solutions for each of the islands to enhance global exploration as specified in Eq. (6):

$${X_{merged}}=\bigcup\limits_{{m=1}}^{M} {\{ {x_{best}},m} \} ,{\text{ }}{x_{best}},m \in {X_m}$$

(6)

In order to prevent the updates from being outdated, communication occurs at some predetermined intervals T_s, during which all processors share their intermediate best solutions, as illustrated in Eq. (7):

$${x_{globalbest}}=\arg \mathop {\hbox{min} }\limits_{m} f({x_{best}},m)$$

(7)

The PDCOA uses interleaved randomization (such as Latin hypercube sampling) to initialize a population in parallel for balanced distribution across the search space. This population is divided into subsets and apportioned to different processors for efficient offloading. The parallel computation of fitness evaluation allows the use of multiple processors, leading to faster computation.

After evaluation, individuals are restructured according to a leader-follower-chaser-chimp hierarchy. Thanks to a dispersed subpopulation update mechanism, an efficient evolution approach enables rapid development. Subpopulation best solutions are periodically exchanged to foster diversity and avoid stagnation while exploitation and exploration are balanced.

PDCOA’s computational efficiency benefits from parallel and distributed attributes, making it suitable for large-scale optimization. It defends against overexploitation and scavenging, robustly aligning the subpopulations to improve convergence speed and maintained performance over complex tasks.

Evolving LSTM through the parallel and distributed chimp optimization algorithm (PDCOA)

LSTM networks are a recurrent neural network architecture now commonly used for time-series prediction, natural language analysis, and speech processing, among other things. The performance level of LSTMs is also related to hyperparameter settings, such as the number of hidden cells, the learning rate, and the dropout rate. However, optimizing such parameters is time-consuming as it involves many configurations that must be gone through with ample search space. Instead, the PDCOA provides a faster solution by leveraging parallelism and distributed computation of this process.

In this context, the PDCOA simultaneously populates a pool of potential LSTM candidates to maximize architectural diversity by using Latin hypercube sampling to fill the hyperparameter space. Each candidate solution is uniquely defined in terms of LSTM hyperparameter combinations. The population is partitioned into categorical groups, with each division being given a specific processor. Each processor will utilize a particular training dataset to assess its subdivided LSTM configurations and determine the suitability levels of the model through evaluation metrics according to the problem context.

Following the fitness evaluation, each processor alters the hyperparameter credentials of its candidates by using the position update mechanism of the PDCOA. This method imitates the chimp neurons’ alpha, subordinates, and beta roles, assisting the search space in exploration and exploitation. As a rule of thumb, each processor integrates the best-performing candidates within their respective subpopulations; hence, information from the best candidate is disseminated throughout the integration. This collective step avoids early solutions and helps the algorithm find optimal or nearly optimal LSTM parameters.

The iterative procedure goes on in a loop until a specific stop condition is fulfilled; for example, a certain level of performance is achieved, or the count of iterations reaches the set maximum. With the help of the PDCOA, the complex task of optimization takes place quickly and is made more efficient, making the PDCOA scalable and capable of dealing with bigger LSTM networks and datasets. Even so, the factors of intermittent communication among the subpopulations provide the robustness of exploration of the hyperparameter space, which decreases the chances of the obtained configurations performing poorly on the unseen data. The whole block diagram of the proposed model is shown in Fig. 2.

Fig. 2

The whole block diagram of the proposed model.

Deploying PDCOA for evolving LSTM networks reduces computation expenses and improves the optimization process’s efficiency. This methodology is favorable in large-scale applications or when time and computational limitations are prevalent.

Experimentation

This section provides the information for the dataset and presents the experiment results.

Well logging

Well logging, also called wireline logging, is essential to the oil and gas industry. It involves finding out the exact geophysical and geological properties of the beds below the surface of a borehole. It investigates the properties of rocks and fluids to plan the search for hydrocarbons and estimate zone porosity, permeability, and fluid saturation and their subsequent management. The most common drilled wells include electric (resistivity, spontaneous potential) and porosity (neutron, density, sonic, and gamma-ray) logs.

In well-log data, missing values may result from equipment malfunction, the nature of the environment, or the covered area of the borehole, and such shortcomings need considerable attention to restore these values using interpolation, imputation, or modeling techniques to make an accurate evaluation of the reservoir and make reliable decisions. Figure 3 is a standard vertical graph that shows the recorded values of these attributes side by side; the x-axis indicates the attribute, and the y-axis indicates the depth.

Fig. 3

Standard vertical graph showing the recorded well-log values using data from the NLOG dataset sampled at mid-depth intervals (500–1500 m).

Dataset

This study relies heavily on the NLOG dataset (https://www.nlog.nl/en), which is available to the public. The data comprises well-log measurements taken from the North Sea Shelf over the Dutch-controlled areas. The logs are available in LAS format⁴², a standard text file with a cover page containing particulars on borehole depth, coordinates, well identification number, the range of depth, property measured, units, and the absence of data. Each file contains a complete record of measurements up to the standard depth band and intervals, usually 10–15 cm apart, separated using tabs, and refer to each row as depth and each column as an attribute.

The dataset measured the most frequently recorded property attributes, such as deviates from standard well-logging mnemonics, such as Delta-T (DT), Gamma-ray (gr), bulk density (bd), and neutron porosity (np). However, these values contained most of the non-value ones. Note that the details of the features are tabulated in Table 2.

Table 2 Details of the features.

For instance, comprehensive datasets show that only 26,000 rows were annotated with all required attributes in 23 million existing ones. A gap is defined as any interval exceeding 0.3 m where data for a property is absent. Just as specific shallow measurements close to the top of the borehole are disregarded due to the presence of seawater, others are deemed to add little value. The average gap size appears to vary by the drilling of separate vertical boreholes, which results in their average being 170 m in thickness for n_p and under 15 m in thickness for g_r. Figure 4 shows the spatial distribution of well-log attributes, e.g., bulk density, sonic measurements, gamma-ray, and neutron porosity.

Fig. 4

Spatial distribution of well-log attributes, e.g., bulk density, sonic measurements, gamma-ray, and neutron porosity in the NLOG dataset.

Test gaps were artificially created at the same depths in selected test wells to compare forecast performance on different target features and various training regimes in these wells. However, wells with scarce training samples were omitted from the analysis to avoid situations of limited training data.

The attributes also exhibited varying probabilities of missing data: 43.1% lost s_m, 24.2% lost g_r, 22.9% lost b_d, and 10.6% lost n_p. The data analysis revealed a weak dependence of one attribute’s missing data on other characteristics, with 84% of the gaps referring to a single property; gaps involving two or more properties were rare. The data do not support a strong argument that geological features determine gaps.

The robustness of the LSTM-PDCOA model was assessed by adding simulated missing data gaps into the well-log time series. The programming of this analysis allows the simulation of well-log time series missing data gaps, applying statistical methodologies of gap size, location, and frequency. Gap size defines the quantitative amount of missing segments, location describes where within the depth profile gaps are situated, and frequency controls how often gaps occur to assess robustness across different data sufficiency levels. Results from the analysis demonstrate enhanced performance compared to baseline methods, achieving 15% lower MSE under the simulated conditions. In particular, synthetic gaps that were 50 ± 150 m were made more significant than the standard sizes to improve the forecasting of complex features, as shown in Fig. 5. The distribution of gaps revealed smaller gaps at deeper levels, and the likelihood of finding them rose as the borehole’s depth increased.

Despite significant variability across individual wells, the NLOG dataset had an average density of 1.6 gaps per km. The synthetic gap generator adopted two gaps per km, which is about the most moderate density per km for the simulation. Regarding regression tasks, the principal loss function was a mean squared error (MSE), which looks at how far the prediction moved up or down compared to the target range. While parameters with a highly distributed range can easily accept an error of 100 points, parameters with minimal ranges should ideally have an error of less than 100. This brings about the need to understand the level of MSE in its MSE context.

Fig. 5

Preprocessing and artificial gaps.

Implementation

Models were executed on the TensorFlow framework using the Python programming language and the Keras API for high-level usages such as deep learning and data handling. The tool utilized in development was Jupyter Notebook, which was used locally on a workstation with an NVIDIA RTX 3090 GPU to speed up the training process. The data and the models were maintained on an internal SSD to minimize time and maximize storage efficiency. The training process included 30 epochs.

The LSTM architecture consisted of 15 input layer nodes, four hidden layers with 4000, 2000, 2000, and 30 nodes, respectively, and an output layer with two nodes. These configurations were determined via sensitivity analysis. The hidden layers use ReLU activation to introduce nonlinearity. In contrast, the output layer employs a linear activation function to predict continuous well-log values, aligning with the task’s regression nature.

The NLOG dataset extracted from the North Sea Shelf (https://www.nlog.nl/en) was filtered considering the following well inclusion criteria: depth greater than 1500 m, maximum gap of 50 m, inclusion to sample ratio greater than or equal to 0.5, and at least 750 m of usable log data per well. After filtering, the dataset was reduced to roughly 12,000 samples across the selected wells. Preprocessing included treating missing values and feature scaling. All the well-log attributes (like DT for sonic, NPHI for neutron porosity, GR for gamma-ray, and RHOB for bulk density) were scaled to a range [0,1] for model training, while denormalizing predictions back to their original values (DT in µs/ft, NPHI in fraction) during evaluation. The dataset was split into training, 70%; validation, 15%; and test, 15%, with stratified random sampling using a predefined split, which equated to approximately 8,400 samples for training, 1,800 samples for validation, and 1,800 samples for testing. This split was done in each well, using stratification for depth (≥ 1500 m) and data missingness patterns (e.g., 43.1% missing sonic data) to ensure each subset is representative of the dataset variability instead of whole wells or complete zonations.

Given that a random seed (42) was applied for reproducibility, each set underwent independent preprocessing to avoid data leakage. This method, as illustrated in Table 3, upholds the integrity of the data while reducing bias.

Table 3 Subsets of dataset.

LSTM-COA⁴⁰, LSTM-DLBMPA (Dynamic Lévy–Brownian marine predator algorithm)⁴¹, LSTM-SBO (snow ablation optimizer)⁴³, LSTM-HBA (honey badger algorithm)⁴⁴, and LSTM-QCOA (Quantum COA)²⁷, as well as standard forecasting models using XGBoost⁴⁵, LightGBM⁴⁶, AdaBoost⁴⁷, and CatBoost⁴⁸ models were built to predict missing values. Table 4 shows the basic parameters and setup values for these optimization techniques.

Table 4 Setting parameters and initial values.

Model evaluation and comparison

LSTM-based models (COA, DLBMPA, SBO, HBA, QCOA, and the proposed PDCOA) were put to the test versus traditional time series models, which are XGBoost, LightGBM, AdaBoost, and CatBoost. The evaluation made use of metrics important for forecasting accuracy and robustness:

Mean Squared Error (MSE): Average of the squared prediction error. Ideally, lower values indicate better predictions. Mean Absolute Error (MAE): Presents how large or small the errors are but ignores the domino effect. R-Squared (R²): Models the relationship, in percentage, of variation that the model explains. Root Mean Squared Error (RMSE): A more intense form of MSE focused on the more significant errors. The Mean Absolute Percentage Error (MAPE), rescaled for easier comprehension, provides an understanding of accuracy⁴⁹. The proposed LSTM-PDCOA model was validated using k-fold cross-validation to improve its robustness and generalizability. We employed a 5-fold cross-validation, splitting our dataset into five equally sized subsets. Four of the five subsets were used for training, while one was used for validation. Every data point was predicted only once. The performance metrics, such as MSE, MAE, R², etc., were averaged across folds to lessen the effects of data splitting on model evaluation. Such estimates are less prone to overfitting and are deemed to be more reliable. In addition, simulation studies were also conducted to examine model performance across varying levels of missing data and noise. The performance difference was statistically tested using Wilcoxon signed-rank tests. The Wilcoxon rank-sum test, or Mann-Whitney U test, is a test applied to two independent samples to ascertain the difference in their population distributions⁵⁰. It tests the hypothesis of whether the population of one group has lower or higher values than the other population by ranking all the provided observations together and calculating the sum of the ranks. This test efficiently completes work even when the data fails to meet the parametric test requirements of normality. It is routinely used for ordinal or continuous data assumed to be non-normally distributed. To compare the different models consistently, the following hypothesis was put forward:

Null hypothesis (H₀)

No difference is expected between the models regarding predictive performance measures such as the MSE value. That is, comparison models will not show significant differences in their predictive abilities across the samples.

Alternative hypothesis (H₁)

The predictive performance measures of the tested models will differ significantly.

Wilcoxon signed rank test was used to assess the difference in the performance metrics across the models. This test was employed in our study because it is a nonparametric test that makes no assumptions about the normal distribution of errors and is appropriate for paired data. The outcome measures showed significant disparities in model performance metrics as indicated using the Wilcoxon Signed Rank Test (p < 0.05). The obtained results are tabulated in Table 5.

Table 5 The obtained results.

The results verify that the LSTM-PDCOA model has the best results in all evaluation metrics and comparatives. In particular, PDCOA’s low error rates are unprecedented with an MSE of 0.0098, MAE of 0.0295, RMSE of 0.0990, and R² value of 0.99 with MAPE at 2.7%. PDCOA performs better with noisy data (e.g. MSE 0.0153 at 20% noise) as well as with missing data (e.g. MSE of 0.0167 at 50% missing data) where it still retains statistical significance (p < 0.05), as validated by the Wilcoxon signed-rank test and the declared performance difference from LSTM-COA, LSTM-DLBMPA, and XGBoost and AdaBoost models. Taylor plot analysis confirms that PDCOA is the most converged model when compared against the rest due to the lower correlation distance and RMSE to the observed data.

The enhanced performance of PDCOA is due to its original implementation of parallel and distributed computing, which optimizes a given task by splitting it into multiple subpopulations that are worked on simultaneously at different nodes. This approach reduces computational time (for example, average search time of 8 m 56s versus 75 m 42s for the Grid Search). In addition, periodic communication between subpopulations maintains a balance between exploration and exploitation, which avoids getting stuck at local optima and improves solutions’ diversity. Compared to sequential optimization methods such as COA, PDCOA is more efficient at scales with large datasets due to its distributed framework, which is also more robust to noise and missing data patterns due to the coordination mechanism. The outcome evaluated in the statistical tests showed that these factors enable PDCOA to outperform other algorithms in optimizing LSTM hypermodels, which resulted in accurate predictions of missing well-log data.

Noise robustness experiment

The simulation examining noise robustness is critical in examining the efficiency of machine learning models following exposure to noise-robust data. Indeed, in such practical situations, when carrying out geophysical surveys or sensor-based applications, data could be inaccurate due to various reasons ranging from environmental to measurement errors. The simulation provided noise to the dataset and tested whether a model could predict accurately in such disturbances. A good model will likely generalize the unseen data effectively and provide more robust predictive utility. This simulation ensures that the model can cope with the imperfections of the data and does not simply overfit the noise in the data. The obtained results are presented in Table 6; Fig. 6.

Table 6 Noise robustness results in lower MSE values, indicating better noise resilience.

Fig. 6

Noise robustness plot.

With the increase in dataset noise level, there is an increase in the error metrics for all the models, indicating the difficulties noise poses on the models. Some models perform well on terrains with clean data. At 0% noise, PDCOA shows the lowest error value of 0.0098, higher than the 0.0105 exhibited by DLBMPA and 0.0102 shown by QCOA. The highest error of 0.0178 was reported for AdaBoost; hence, it has been noted that it is inefficient with clean data, unlike the other models. As the noise increased to 5%, 10%, and 20%, all the models recorded a gradual rise in error, although AdaBoost recorded the worst performance at 0.0201 at 5% noise. While the HBA, LightGBM, XGBoost, and LSTM-PDCOA models all showed a consistent trend of increasing errors with higher noise levels, the HBA model suffered the most in terms of performance compared to the other models. LightGBM and XGBoost showed some degree of robustness, demonstrating much lower increases in error. XGBoost’s MSE increased from 0.045 to 0.055, while LightGBM’s MSE rose from 0.048 to 0.058 with 20% noise added. Comparatively, LSTM-PDCOA suffered an increase from 0.0342 to 0.044, which is still relevant, indicating they did not deteriorate as much as initially claimed.

Missing data patterns

The simulation of missing data patterns is essential for assessing models’ performance in the case of incomplete data, which is quite prevalent in practice. Sensor malfunctions, human error, or problems in data transmission in applications like healthcare, finance, and geophysical studies might create gaps in data. By purposefully missing certain sections of the data set, we can test how these models perform in practice, be it through imputation of missing values or other available methods. This simulation assists in determining what models would produce accurate outputs in situations where information inputs are deficient. It also allows an understanding of how well the model performs if certain information is missing, thereby increasing its reliability, as data incompleteness will persist. The obtained results are presented in Table 7; Fig. 7.

Table 7 Missing data patterns with different missing data rates (10%, 20%, 30%, and 50%).

Fig. 7

Plot of missing data patterns.

The increase in the missing data ratio leads to a marked increase in the error of all the models. All models show signs of the presence of missing data. The presence of missing data tends to lower their performances. For 10% missing data, the rates of models are entirely satisfactory, with PDCOA achieving the lowest error rate of 0.00990, followed by QCOA at 0.0105 and DLBMPA at 0.01102 in error rates. However, AdaBoost, which has the highest error rate of 0.0180, appears to suffer more from the missing data rates in its models. At 20% missing data, the situation becomes worse for all models. The error is noticeably higher in all models; however, PDCOA retains the lead with a 0.012305 error. AdaBoost reaches the worst of the standards at 0.0202, which is the highest error. As the model aims for a missing data rate of 30%, PDCOA still provides the best performance with 0.0145. However, the gap between PDCOA and the remaining models is considerably smaller. The weakest model is still AdaBoost, which performs this time at 0.0229. Other models, such as SBO and LightGBM, also take a sizeable hit. Finally, this is a staggering increase for a model with 50% missing data. All models exhibit tremendous steep increases as well as their error rates. PDCOA maintained its lead with the lowest error at 0.01671 but suffered considerably. AdaBoost, however, turns out to be the victor in the worst case—this model maintains its lead over the others, demonstrating the highest error out of all the models aimed at 0.0251. This analysis makes it clear that PDCOA is the model most resistant to missing data, while AdaBoost suffers without a doubt as the level of missing data increases.

Taylor plot analysis

A Taylor plot is a functional representation to examine how well different models perform, mainly how much they differ from real-world data concerning correlation, standard deviation, and RMSE metrics. This subsection intends to end the discussion by demonstrating the activity, the Taylor plot, and model intrusion detection activity through various machine learning models, including scripts over noise robustness and patterns of missing data.

The Taylor plot integrates the standard deviations of a collective of models, the model correlation of such data with the real observable data, and the RMSE of that model into a single figure. It enables the assessment of each model’s performance when aligned with the perfect model: one with a perfect score of 1 for correlation, an equal score for standard correlation, and the lowest score for RMSE.

To calculate the Taylor plot for our models, which include but are not limited to LSTM-COA, LSTM-DLBMPA, LSTM-SBO, LSTM-HBA, LSTM-QCOA, XGBoost, LightGBM, AdaBoost, CatBoost, and PDCOA, calculations of the correlation, standard deviation, and the RMSE were performed on all models during the noise robustness and the missing data patterns simulation. These will be used to construct figures that will illustrate the relative performance of different models on simulated tasks. Figure 8 shows the Tylor plot for comparison models.

Fig. 8

Tylor plot for comparison models.

The Taylor plot complements the understanding of models’ proficiency by illustrating the relationship between a given model’s standard deviation, correlation coefficient, root mean square error (RMSE), and observation data. The ideal model is plotted at the coordinate (1, 0) in the plot space, indicating a perfect correlation with the matching standard deviation and zero RMSE. Models that are closer to this reference point are regarded as performing better.

From the results of PDCOA, the proposed model is the one closest to the observed point, which displays strength toward missing data patterns and noise. Models such as LSTM-COA and LSTM-DLBMPA have also shown strong correlation and competitive standard deviation. Still, they are, to some extent, lagging in terms of RMSE, which indicates their predictions are relatively more off. Also, XGBoost and LightGBM, which are conventional boosting algorithms, have moderately performed with low correlations and slightly high deviations due to their inability to deal with complex temporal dependencies inherent in the dataset.

Computational complexity

Computational complexity is an important consideration, especially in the advanced areas and towards the field applications of any machine learning model. Figure 9 presents the computational complexity of the analyzed models in terms of FLOPS (floating point operations per second), the number of parameters, and training time.

Fig. 9

Computational complexity.

Figure 9 illustrates the performance of the models LSTM, LSTM-PDCOA, LSTM COA, LSTM DLBMPA, LSTM HBA, LSTM SBO, and LSTM GOLSCOA in terms of FLOPS, Number of Parameters, and Training Time. The ranking indicates that LSTM is the best overall, and LSTM-PDCOA is second. Their ranking is justified with LSTM having the lowest cost in computation 7.3 × 10³ M FLOPS, 10.22 × 10³ parameters, and 491 s training time, while LSTM-PDCOA follows closely with 7.46 × 10³ M FLOPS, 10.23 × 10³ parameters, and 536 s. The rest of the models were ordered by minimum training time and show greater stratifying complexity, with LSTM-GOLSCOA having the worst 8.95 × 10³ M FLOPS, 10.08 × 10³ parameters, and 868 s training time, meaning that they have lower efficiency but higher accuracy in predictions.

The results portray the situation of inverse accuracy and resource usage, meaning that less efficient models like LSTM GOLSDCO can provide higher accuracy. In contrast, resource-efficient models, such as LSTM and LSTM-PDCOAD, offer more excellent value for time-starved and resource-fatigued situations. This aids in elaborating the strategy for model selection belonging to a specific task defining a limit on performance and cost efficiency, concentrating more on resource allocation which can be optimized with LSTM and LSTMPDCOA. In contrast, LSTMGOLDCOA can be used when efficiency is not as important.

Comparison with traditional methods

To assess the LSTM hyperparameter optimization capabilities of the PDCOA, we benchmarked its performance concerning two LSTM optimization methods that are popular in practice:

Grid search: the ‘brute force’ method of going through the hyperparameter space to find the best value⁵¹.

Keras tuner: An elegant library for hyperparameter tuning that implements some intelligent algorithms (random search, hyperband, etc.) to find relevant hyperparameters⁵².

For both baseline methods, we allowed the number of hidden layers, learning rate, dropout rate, and the batch size of the LSTMs to be varied across the same dataset and evaluation settings. Table 8 shows the results.

Table 8 The comparison between PDCOA and traditional methods.

During the comparison of PDCOA with grid search and Keras Tuner, it was noted that PDCOA had reasonable computational costs and high levels of prediction accuracy. Grid search has delivered reasonable results, but its costs in terms of systems resources were prohibitive due to the nature of its search approach. Keras Tuner improved the results a little but could not equal the performance of the PDCOA system because it did not address many complex interactions of the hyperparameters.

The results indicate that PDCOA advances traditional approaches in terms of predictive performance and significantly reduces the computational cost. PDCOA’s parallel and distributed structure enables it to dramatically reduce the time required to search the hyperparameter space, which is helpful for problems involving high dimensionality, such as optimizing LSTM networks.

Experimental results

This study aimed to evaluate the performance of the seven models in predicting the missing well-log data for L17-02. The findings, illustrated in Figs. 10 and 11 highlight some considerable differences in the model’s performance.

Fig. 10

The prediction Well log missing value.

Fig. 11

L17-02’s missing log data.

The LSTM-PDCOA model registered the best porosity predictions with the lowest means on both MSE and MAPE. The actual measurements were also very close to the model predictions, showing the high reliability of the model. On the other hand, the LSTM, LSTM-COA, LSTM-DLBMPA, and LSTM-SBO models showed more significant errors and saw more fluctuation in their neutron porous models. Due to robust models, all LSTM-PDCOA again showed the best competition with the least error metrics for the GR value case. Other models were weaker in predictions with higher error metrics due to the instability of the models when bulk density varied. In all instances where stable grids were required rather than fluctuating, LSTM-PDCOA consistently gave values the closest to the conditions warranted. The plots illustrated each model’s performance in that the black dashed line represented the ground truth while the various colored dashed lines represented the models’ forecasts. Being closer to the black line represented better model accuracy, while the black line was ideal model accuracy, which can never be attained. In the case of LSTM-PDCOA, which consistently made robust models, accurate predictions for the missing well-log data were made.

Such results mean that LSTM-PDCOA can be used for high-accuracy, well-logged data prediction and has broader potential applications in the petroleum industry. Future work may focus on improving the accuracy of the models by adding more features or improving error measures.

Error distribution analysis

To assess the performance of the LSTM-PDCOA model concerning the prediction error distribution, we evaluated residuals (predicted vs. actual values) associated with the test set. In particular, we analyzed the residuals associated with sonic (DT) predictions because DT is a critical well log attribute with a substantial amount of missing data (43.1%). The test set includes approximately 1,800 samples (15% of the filtered NLOG dataset, which contains ~ 12,000 samples) from wells with depths ≥ 1500 m.

Residuals for DT predictions were calculated by taking the difference between actual and predicted values on the test set. To determine whether the errors adhere to a normal distribution, we used the Shapiro-Wilk’s test, one of normality’s best renowned statistical tests. The test produced a p-value of 0.03 (p < 0.05), confirming that the assumption of normality can be rejected at a significance level of 5%. Hence, the disclosure is determined to be not normally distributed. Further examination indicates the presence of minor positive skewness (0.4) and kurtosis (0.6), which means these values suggest a distribution slightly above the average. To illustrate this finding, we created a histogram of the residuals, which has been displayed in Fig. 12.

Fig. 12

Histogram of residuals for LSTM-PDCOA on test Set (DT).

This histogram was constructed using 50 bins and refutes the Normal distribution claim as it has a slight positive skew with a heavier tail solely on the positive side. This implies that the model overpredicts DT values and does not underpredict them. This non-normality may be explained in the heterogeneous NLOG dataset containing wells from different geological formations within the North Sea Shelf, including the effects of missing data patterns (43.1% of sonic data was unavailable). The presence of geological variability, for example, changes in lithology or formation characteristics, can produce non-uniform error patterns. At the same time, the absence of data adds extra difficulty in prediction certainty at various depth intervals and across different wells.

However, the fact that cross-validation test scores have high R-squared values signals that most of the features included alongside the target yield the predicted outcome suggests that the LSTM-PDCOA model reaches a relative high R² value of 0.92 and MSE equal to 0.0342 on the test set while maintaining accuracy prompts us to focus on test data, confirming that the model has sustained predictive accuracy. As mentioned before, non-uniform error distribution does not impact the model’s effectiveness; instead, it serves as constructive criticism for relying on error mitigation strategies or incorporating geological features in model changes. This scrutiny meets the study’s intent to reconstruct logged well data marked by imprecision.

Discussion

The main goal of this work is to assess and compare the performance of deep learning models, namely LSTMs, LSTM-COA, LSTM-DLBMPA, LSTM-SBO, LSTM-HBA, LSTM-GOLSCOA, and LSTM-PDCOA, concerning three metrics: FLOPS (Floating Point Operations per Second), number of parameters, and training time. The primary aim was to evaluate the extent to which the model is computationally efficient, complex, and feasible for real-world applications.

Computational efficiency and complexity

Significant discrepancies exist in the computational efficiency of the data models, and the FLOPS metric showcased this variation exceptionally well. For instance, LSTM-GOLSCOA and LSTM-DLBMPA had better FLOPS, which, in this case, implied that this model was more computationally demanding. These results are anticipated because of higher model parameterization in these models, which tend to have greater detail in data-capturing processes to improve predictions. However, higher FLOPS values in the models also indicate higher computational costs, which may be challenging in applications with limited resources, especially in real-time requirements. LSTM-PDCOA, on the contrary, had the lowest FLOPS in the analysis, representing its efficiency well. This model is ideal for scenarios where the cost of computing resources must be minimal, particularly in low-powered or cloud computing models.

The number of parameters, proportionately the most influential in determining the complexity of a model, followed a similar trend. For instance, the models LSTM-GOLSCOA and LSTM-DLBMPA were found to have many parameters, which is quite logical, as models with many parameters often perform complicated tasks and improve their predictive power. As a general rule, the more parameters, the better the model’s performance, but this comes with increased resource and memory requirements, which may not be ideal in cases where they are in short supply. Suitable models with fewer parameters include LSTM and LSTM-PDCOA, which require low memory and execute faster but risk rigidity or low predictive accuracy to some extent.

Training time and model efficiency

The practical applicability of the model concerning time is squarely impacted by the training time. LSTM-GOLSCOA has a long training time, which aligns with expectations, as its high parametric balance and higher computing cost will demand such time. Such a training duration is acceptable, provided the additional complexity translates into better prediction accuracy or performance measures. Nevertheless, a long training duration may be too restrictive for several real-world applications.

On the other hand, the realistic training duration for LSTM-PDCOA and LSTM-COA was much shorter, making them more feasible options in situations that require rapid turnaround. LSTM-PDCOA, for instance, could perform similarly to the other models but in much less time. This trade-off of computational resources and training speed further makes LSTM-PDCOA a potential candidate in cases that require rapid results, such as operational ones, or when working with time-varying data.

Considerations for model selection

It is abundantly clear from this investigation that the model, if not pragmatic functionality, is a significant hurdle to overcome. More realistic models, however, like LSTM-GOLSCOA and LSTM-DLBMPA, would have increased accuracy because of their many parameters, but this is accompanied by long training times and many FLOPS. This is a considerable trade-off that must be considered when selecting models for practical applications, considering the shortage of resources like time or computation power.

In a situation where the time to implement the model and the required resources are less, simpler models like LSTM-PDCOA and LSTM-COA, which are less complex and take less time to train, can be considered ideal candidates. Indeed, these models are likely to underperform in capturing the entire data structure compared to more sophisticated models; however, they provide a better option when the priority is on speed, and the usage of resources is limited.

Practical implications

The results of this study should enable practitioners to make a prudent decision about adopting the model as per the peculiar needs of the intended application. For example, if the computational power is high and the level of accuracy needed is very high, advanced models such as the LSTM-GOLSCOA or LSTM-DLBMPA will be the most suitable models. On the other hand, if real-time response or lesser consumption of resources becomes critical, the models should be LSTM-PDCOA or LSTM-COA, as they integrate good performance with less cost.

Limitations of the study

In as much as the research offers a significant understanding of the operationalization of different machine learning models, the study has some inherent limitations that need to be stated. Firstly, the models under test were trained on a fixed data set and may not necessarily perform well on other forms of data. Future work could involve applying these models to various databases in different domains to evaluate the model’s robustness and other characteristics.

Secondly, it is unclear how hyperparameter tuning was conducted, which often significantly impacts model performance. Adjusting hyperparameters, including the learning rate and batch size, can be beneficial and even necessary for the more complex models, such as LSTM-GOLSCOA and LSTM-DLBMPA.

Another concern is the overreliance on FLOPS, the number of parameters, and training time as performance measure metrics. Though it is significant to have these measures for computational efficiency and complexity assessment, it is also worth considering other measures, including model interpretability or model scalability, in relative applications in upcoming studies. Therefore, it is recommended that more experiments are carried out using diverse hardware configurations to test these models in various environments.

Future directions research

In the future, we might seek ways of optimizing the trade-offs between model complexity, computational cost, and time spent training a model. As an example, this would entail the development of hybrid models that combine the strengths of both complex and efficient ones. Furthermore, investigation into more advanced forms of regularization, such as pruning and parameter sharing, might reduce the number of parameters of the model but still allow it to perform at the highest levels. Furthermore, as the development of hardware resources continues, especially with the development of novel accelerators such as TPUs and FPGAs, the efficiency of these models may be assessed afresh to determine whether more complex and resource-demanding models may be applicable in practice.

Despite these restrictions, this paper offers followers of deep learning a rich contextual review of different deep learning models and the appropriate balancing act amongst model complexity, training time, and computational efficiency. These findings can be translated and adapted into the practical applications of the models in many fields.

Conclusion

This research developed and presented a new approach based on machine learning techniques to solve the problem of missing or incomplete well-log data, which is one of the significant problems in the oil and gas sector. We developed a deep LSTM network with the PDCOA to reconstruct the missing well-log data efficiently. We illustrated its usefulness in several simulated studies and compared it with contemporary algorithms. After looking at the framework’s results, it was found that LSTM-PDCOA worked better than the LSTM-COA and LSTM-DLBMPA networks, as well as some more traditional machine learning methods like XGBoost, LightGBM, AdaBoost, and CatBoost. More importantly, this model reduced computational complexity, increased prediction accuracy, and reduced training times, making it an asset in geotechnical analysis and hydrocarbon volume prediction.

When it comes to these results, a number of these limitations were recognized and should be resolved in future work. One of these limitations was the reliance on excellent and complete well-log datasets, which are not necessarily available in practical application. In addition, diversified datasets from other countries and regions or geological models could have further boosted the model’s performance. Also, although the objective of the work was to reconstruct missing data, testing the model’s ability to offer real-time updates and decision-making during actual oil-expanding schemes would be a good next step. Another avenue to pursue is enhancing the PDCOA algorithm to achieve even faster convergence and better scalability to larger datasets, enabling its application in more complex geologic and seismic data analysis.

In conclusion, the findings indicate that the LSTM-PDCOA method can significantly enhance the precision of oil exploration activities. If it gets even better and can be used with other types of geologic data, it will change how missing well log data is restored. This will help people exploring for and extracting hydrocarbons make better decisions. Future directions may emphasize overcoming the existing limitations, increasing the applicability of the model, and seeking new optimization algorithms to improve the model’s robustness and accuracy of the forecasts.