Development of a Control-Aware Digital Twin Framework for Real-Time Monitoring and Optimization of Oil and Gas Production Systems

Summary

This study presents a control-aware digital twin framework for monitoring and optimization of oil and gas systems using machine learning and physics-informed models, evaluated on simulated and benchmark datasets.

Abstract

A control-aware digital twin framework for tracking and improving oil and gas production systems is suggested in this paper. The method combines spatiotemporal graph neural networks, neural 4D-Var data assimilation, reinforcement learning-based control, and physics-informed neural operators (FNO and Neural operator model). Benchmark cyber-physical system datasets (SWaT and WADI) and a simulated oil and gas dataset are used to assess the framework. The results, which were confirmed by several independent runs using average measures, demonstrate enhanced performance in state estimation, anomaly detection, and control optimization. However, the assessment is restricted to benchmark and simulated datasets; additional validation using actual industry data is needed to verify practical applicability.

Introduction

The need for safe and effective oil and gas transportation networks has increased due to the world's growing energy demand. However, sustaining pipeline reliability is severely hampered by deteriorating infrastructure and growing system complexity. In this regard, digital twin (DT) technology has drawn interest as a data-driven lifecycle management strategy that permits risk assessment, predictive maintenance, and system optimization in industrial settings^1,2,3,4,5. From reactive maintenance to intelligent, data-driven decision-making, digital transformation in industrial systems has increased anomaly detection, failure prediction, and system reliability^6,7,8. DT applications in process optimization, predictive maintenance, and smart manufacturing have been the subject of numerous research. For example, DT-based models have been suggested for predictive failure analysis¹⁰ and green performance evaluation¹¹, and deep learning-based production optimization. DTs have been used in the oil and gas industry to improve monitoring and anomaly identification through system modeling, performance assessment, and multi-sensor data integration^12,13,14,15.

Despite these developments, the majority of current DT techniques concentrate on discrete features like monitoring, modeling, or anomaly detection, with little integration of real-time synchronization and control. Many systems lack bidirectional interaction and adaptive learning capabilities, functioning as digital shadows with unidirectional data flow. In oil and gas systems, where dynamic conditions and intricate fault patterns necessitate ongoing learning and control, this constraint is especially important. Additionally, real-time closed-loop control in complicated industrial processes is not supported by current cyber-physical system techniques, which are mostly limited to offline analysis¹⁶. Additionally, in order to describe distributed sensor interactions, current approaches mostly rely on traditional machine learning and filtering techniques, with little usage of sophisticated models like spatio-temporal graph neural networks. Additionally, there is a lack of research on the use of operator learning techniques to capture nonlinear physical dynamics, especially in oil and gas production systems¹⁷. Because of this, existing DT systems are unable to incorporate real-time data assimilation and adaptive control while modeling intricate spatiotemporal connections.

Because they mimic industrial control settings, benchmark datasets like SWaT and WADI are frequently used to validate DT-based techniques in cyber-physical systems¹⁸. Additionally, DTs provide more general benefits, such as increased productivity, better knowledge exchange, and safe data transfer throughout industrial processes^19,20. Particularly in high-risk industries like oil and gas^21,22,23, their potential in safety management, real-time monitoring, and industrial automation is becoming more widely acknowledged. Nevertheless, more complete and integrated solutions that integrate modeling, monitoring, and control capabilities are still needed for practical deployment. By demonstrating the uses and benefits of DTs in improving efficiency in operations and the way decisions are made, evaluation studies such as those cited in^24,25 have laid the foundation. Table 1 shows the comparison of existing methods about oil and gas manufacture and control management^26,27,28,29. Due to the high operational risks and complicated dynamics of oil and gas production systems, traditional maintenance methods are frequently inadequate for prompt anomaly identification and reaction^30,31,32,33. Even though safety management and remote monitoring have improved recently, intelligent systems that can continuously sense, learn, and control in real time are still required.

This paper suggests a control-aware digital twin architecture that combines spatiotemporal graph neural networks, neural 4D-Var data assimilation, physics-informed neural operators, and reinforcement of learning-based control to overcome these difficulties. A single pipeline for data collection, state estimation, anomaly detection, and control optimization is established by the suggested system. While spatiotemporal graph neural networks record interactions among distributed sensors, physics-informed neural operators, such as Fourier Neural Operator and Deep Operator Network (Neural operator model), are used to represent complex nonlinear system dynamics. Reinforcement learning allows for adaptive, closed-loop control, while neural data assimilation improves state estimate accuracy. The suggested method's generalizability across industrial settings is demonstrated by evaluating it using benchmark cyber-physical system datasets (SWaT and WADI) and simulated oil and gas data. This work overcomes major shortcomings of current DT systems and offers a scalable and adaptable solution for real-time monitoring and optimization in oil and gas production systems by combining modeling, learning, and control into a unified framework.

Protocol

This study does not involve any human participants or animal subjects. The datasets used (SWaT and WADI) are publicly available benchmark datasets, and no sensitive or personal data is involved.

The proposed approach establishes a control-conscious digital twin enabling the incorporation of cutting-edge deep learning algorithms, data assimilation, and continuous evaluation and optimization of the oil and gas extraction systems and control. First, real-time multi-modal sensor data from geographically distributed production infrastructure are gathered and modeled using a graph structure that defines connectivity and interactions between production components. Then, physics-informed neural operators are leveraged for the construction of operator-level approximations of the governing equations governing non-linear multiphase flow, facilitating scalable simulation within the digital twin. In the research, the term ‘Neural operator model’ has been used to refer to the Deep Operator Network. In order to incorporate spatial and temporal dependencies among geographically distributed production assets, spatio-temporal graph neural networks are trained using within the physics-informed representations. A differentiable data assimilation approach leveraging neural 4DVAR is applied for maintaining the digital twin simulation predictions consistent and synchronized with real-time monitoring data, even with noisy and spatially irregular measurements. On top of the real-time synchronized digital twin, a reinforcement learning-driven optimization tool is trained using for real-time control. Finally, the digital twin remains equipped for continuous updating of models and adapted scalability over a prolonged duration based on continuous refreshes from new available measurements.

The proposed digital twin architecture enhances the conventional digital twin architecture by adding intelligence, adaptability, and control-awareness to it. The entire pipeline is composed of four modules: (i) data acquisition/graph construction, (ii) state estimation by means of neural operators and ST-GNN, (iii) anomaly detection by means of residual-based analysis, and (iv) reinforcement learning-based control.

Figure 1: Overall Control-Aware Digital-Twin Architecture. Anomaly detection, spatiotemporal graph neural networks, neural 4D-Var synchronization, physics-informed neural operators, and reinforcement learning-based closed-loop control for oil and gas production systems are all integrated into the proposed control-aware digital twin model. Please click here to view a larger version of this figure.

Figure 1 represents the functional process of an oil and gas production control-aware digital twin as a closed-loop cyber-physical system. The functional process begins at the oil and gas production system level, in which different sensors distributed on wells, pipelines, valves, and processing units’ capture data on pressure, rate of flow, temperature, and controls at an instant in time. The various sensors capture data in various formats, and the data is sent to the digital twin, and this constructs the foundation of organization awareness at the instant.

The proposed methodology involves subsequent steps:

Data collection

Real-time information from sensors regarding pressure, flow rate, temperature, and control inputs is taken from oil and gas production environments. Other available benchmark datasets like SWAT and WADI are used for robustness and generalization capabilities. To ensure reproducibility, each step of the suggested methodology is specified using technical details.

[Please refer to Supplementary File (S1) for Dataset Generation and Simulation.]

Two public benchmark industrial control system datasets from the research domain for industrial control systems are used to prove the capability of the conceptual digital twin with respect to complex and high-dimensional time-series data as well as unusual events: The Secure Water Treatment (SWaT), and the Water Distribution (WADI) benchmark datasets³⁴. Both datasets are derived from realistic cyber physical testbeds and include multivariate sensor and actuator operation data for both normal and anomalous/attack scenarios triggered on such systems. By using SWaT and WADI datasets, it is possible to do cross-domain testing and prove generalizability beyond cyber physical testbeds to broader distributed industrial systems with location and time-related dependencies. The Dataset Description is displayed in Table 2.

The proposed research makes use of three different data sets for the extensive evaluation of the developed digital twin methodology shown in Table 2. The base data set is comprised of simulated data relating to the production organization in oil and gas manufacturing and is intended to reproduce real operational measurements sensed by the system, as well as control and/or fault conditions. Furthermore, the Secure Water Treatment (SWaT) data set is introduced as an example of a reduced-scale physical system, comprising measurements from 51 sensors and actuators across the course of about 11 days, including both normal and artificially created anomalies and attacks. The data set is widely considered as the standard benchmark for research. Another set is the Water Distribution (WADI) data set, whose measurements capture the behavior of the reduced-scale water distribution system comprising about 123 variables over the course of 16 days including both different attacks and normal operating conditions. The data set contains a total of 15 different attacks. The two data sets above revolve mostly around high-dimensional series over time and spatial relationships as in industrial control systems.

Data preprocessing

Before model training, all datasets undergo a thorough pretreatment procedure to guarantee data quality, consistency, and reproducibility. Missing value management, noise filtering, normalization, and feature scaling are among the preprocessing procedures. The detailed preprocessing Pipeline is described in Supplementary File (S2).

System graph construction

In the proposed methodology, the physical production system for oil and gas is represented in the form of a graph, which serves as a mathematical structure for the description of spatial correlations between assets together with the dynamics of the state of the system over time (refer to the Supplementary File (S3) for System graph Construction Formulations).

Physics learning using neural operators

For the correct representation of the nonlinear and multidimensional characteristics of multiphase flow systems, the proposed approach uses physics-informed operators in the form of Fourier Neural Operators (FNO) and Neural operator model. Unlike the traditional method of representing neural networks as pointwise mappings, the concept of operators allows for generalization in terms of different boundary conditions. This feature makes this method more appropriate for multiphase flow systems used in the production of oil and gas since the physical phenomena occur continuously. Please refer to Supplementary File (S4) for neural operators’ mathematical equations.

To further evaluate the validity of the physics-informed neural operators, additional experiments were performed to assess their capacity to approximate nonlinear operator mappings under different flow states and control inputs. The results show that the operators can accurately approximate the mapping of the nonlinear operator, thereby validating the operators’ capacity to accurately model the dynamics of the system, especially in the presence of high dimensionality, at a lower computational cost than other solvers. Quantitative evaluation of operator learning accuracy is performed using mean squared error between predicted and simulated system states, demonstrating consistent approximation performance across different flow conditions.

Figure 2: Workflow of the physics-informed neural operator framework. An example of the physics-informed neural operator workflow demonstrates how control inputs, sensor data, and boundary conditions are used to learn nonlinear multiphase flow dynamics and produce quick, operator-level system predictions. Please click here to view a larger version of this figure.

In the PINO working process presented in Figure 2, the learning process of physically complex nonlinear multiphase flows in oil and gas processes takes place in a continuous manner. This starts at the input layer where the model receives heterogeneous data about the physically evolving system. This input takes the form of boundary conditions, ranging from pressure, temperature, and velocity, as well as real-time data from sensors positioned within the oil and gas system, as well as control variables like the opening of valves, speed of pumps, or opening of chokes. All these inputs define the operation of the physically evolving system.

The key component within the workflow is that of the PINO, realized through FNO and Neural operator model. The proposed method is different and distinct in the sense that it uses neural operators to learn operators between function spaces, as opposed to traditional neural networks, which tend to focus on pointwise mapping between inputs and outputs. At this level, FNO is able to capture the global spatial relationships using the Fourier domain, and Neural operator model is able to learn operator representations using both branch and trunk networks. The learned model is guided by physical constraints to ensure the predicted dynamics behave as expected in the multiphase flow. The outcome of the neural operator takes the form of the continuous field solution of the important physical variables like pressure, flow rates, and phase distribution for the whole production network. Moreover, the output values are accurate for the whole field, including the wells, pipelines, and other connected units, rather than being measured at specific points like sensors. This way, the output of the neural operator provides the capability to rapidly produce highly accurate simulations that can approximate the solution to partial differential equations without necessarily solving them. This solution acts as the fast solution in the digital twin.

The neural operator learning process takes function-level and system-level features, which are extractions from oil and gas production processes. The features consist largely of the physical state variables for oil and gas production, which are continuous fields, temperature, flow rate, pressure, and phase fractions for oil, gas, and water. The features also include boundary conditions, which involve inlet pressures, flow rates, and valve positions. Control inputs, which involve positions for choke valves and pump speed, are also considered for predicting responses under different operational scenarios. The Neural operator model neural operator receives the physical features and parameters for control, which are processed by the trunk and branch networks. The features provide sufficient information for neural operators to predict accurate operator relationships that depend on both physical processes and operational factors for fast digital twin simulations.

Fourier neural operator (FNO)

In this research, a Fourier Neural Operator (FNO)³⁶ is used to effectively estimate the operator mappings that describe the nonlinear multiphase flow dynamics of oil and gas production systems. Contrary to traditional neural networks, which approximate input-output mappings for a specific set of points, the proposed FNO approximates mappings between function spaces, thereby enabling generalization over various scales and configurations of a system. In this case, FNO works by moving the system state representations from the original space to the Fourier domain, where the spectral convolutions can be applied effectively for the purpose of capturing the global spatial dependencies existing within the systems. In this case, the input state representation is projected into a higher-dimensional latent space, and the transforms and activations are applied iteratively using Fourier transforms, spectral kernels, and activations. Please refer Supplementary File (S4.1) for Fourier neural operator mathematical evaluations.

Neural operator model

Neural operator model³⁷ acts as an alternative approach within the neural operator process for approximating a nonlinear physical operator that defines oil and gas production dynamics. The detailed description of Neural operator model is described in Supplementary File (S4.2) (Refer Supplementary File S10 for Algorithm 1: Physics Learning with Neural Operators for Digital Twin Algorithm).

The proposed approach of using neural operators, therefore, commences with gathering paired examples of inputs and outputs, wherein the inputs are examples of system conditions in the form of system boundary values, control variables, or physical characteristics. The neural operator is trained in an iterative phase involving predicting responses to an input function with subsequent improvement of deviations between predictions and actual responses using a mean squared error function. This approach leads to optimization based on gradients, whereby a neural operator discovers systemic physics operators that define system behavior. The trained system can thus efficiently predict system responses to different input conditions.

Spatio-temporal system modeling

In the proposed digital twin architecture³⁸, the oil and gas production process has been represented by a spatio-temporal graph. This has been done to conceptualize the connectivity as well as the time-evolution process of the distributed production resources.

The oil and gas production process inherently has a distributed structure in terms of wells, pipeline infrastructure, and sensors. To efficiently capture spatial/temporal relationships for such a process, the digital twin approach developed in this paper makes use of spatio-temporal graph neural networks (ST-GNN). In this approach, system entities are represented in terms of graph nodes, and system spatial/functional relationships are denoted by graph edges. Temporal relationships are expressed in terms of windowing functions in order to efficiently predict system state transitions and anomalies. Please refer Supplementary File (S5) for detailed description of Spatio-Temporal Graph Neural Network (ST-GNN).

Digital twin synchronization using Neural 4D VAR

Digital twin synchronization can be defined as continuous synchronization of the virtual twin with an ever-changing physical oil and gas production environment based upon real-time sensor readings. In practice, digital twins make predictions about states in a system, such as pressure, flow rate, and temperature, based upon an understanding of physical system dynamics. Nevertheless, due to uncertainties and sensor noise, an inevitable discrepancy arises between predictions and measurements. For this reason, in order to synchronize predictions with actual measurements, another method known as ‘data assimilation’ in this work it introduced to make predictions self-consistent with physical measurements. To ensure synchronization of the physical process and the digital twin during noisy and sparse sensor observations, a neural 4D-Var data assimilation method is incorporated. The added component aims at reducing the gap between sensor measurements and model predictions within an assimilation window. Utilizing differentiability, the neural 4D-Var method provides end-to-end learning and allows the digital twin state to be corrected online (Refer to Supplementary File (S6) for Neural 4D-Var Data Assimilation, Supplementary File (S10) for Algorithm 2: Digital Twin Synchronization Using Neural 4D-Var and refer to Supplementary File (S11) for hyper parameters and training details).

Anomaly detection and diagnosis

In the simulated oil and gas data set, anomalies were injected by introducing controlled perturbations like pressure drops exceeding 15%, abnormal flow spikes, and valve malfunctions. Data points that exceeded a set threshold value or deviated from a baseline statistical distribution are identified as anomalies. For the mathematical formulation of Anomaly Detection Framework (Supplementary File, S7).

Closed-loop optimization and control

In the proposed system, closed-loop control can be achieved by incorporating the harmonious DT and the reinforcement learning module for optimization. In the part of digital twins, the digital twin predicts the future behavior of the oil and gas production process under various conditions such as the positioning of different valves, pump rates, or flow control. This allows the development of a safety area where the control agent can evaluate the result of desired actions without having them materialized in the real-world process. Thus, by exploiting the forecasts offered by the digital twin, the reinforcement learning module can optimize operating goals, including the efficiency of the process, energy use, and process stability, without being limited by process constraints. Please refer to Supplementary File (S8) for Closed-loop Optimization and Control mathematical description.

Continuous learning and adaptation

Industrial systems face non-stationary environments with aging equipment, time-varying operating conditions, and unforeseen disturbances. To design an approach for this, the digital twin model proposed includes an ever-learning component that upgrades neural operators, graph models, and policies based on incremental learning. Upon observing major variations and unidentified patterns, learning is facilitated by incorporating new data and adjusting parameters associated with the digital twin model.

Continuous learning is also improving the robustness of both control policies and detection of anomalies since it helps the system identify new or novel patterns of behavior. Feedback from control result anomalies is applied to adjust models and optimize a reinforcement learning policy that will ensure effective decision-making processes as the system continues to change and evolve. This adaptive learning process changes a digital twin from being a passive observer to an intelligent system that can sustain real-time optimization processes of complex oil and gas production systems over an extended period using a technological infrastructure environment that is rigorous and effective. In Table 3, shows the model architecture and hyperparameters.

Please refer to Supplementary File (S9) for Continuous Learning Adaptation mathematical formulation.

The data preprocessing module is responsible for transforming raw sensor data into structured and normalized inputs suitable for downstream learning modules. Specifically, it aggregates historical sensor measurements, removes noise, and standardizes features to ensure consistent data quality. The processed data, along with boundary conditions and control inputs, is then provided as input to the physics-informed neural operator module (FNO/Neural operator model), which learns the underlying nonlinear physical relationships and generates a continuous system state estimate. This estimated system state is subsequently utilized by the control optimization module to update control strategies in a dynamic manner. The optimized control outputs are then forwarded to the decision-making module, which interprets the results and generates actionable decisions for system operation. A feedback mechanism is incorporated to continuously update the system based on newly observed data, thereby improving adaptability and performance over time. To enhance clarity and reproducibility, a system-level architecture is presented to illustrate the interaction among the modules. The proposed system consists of four main components, namely data preprocessing, physics-informed neural operator modeling, control optimization, and decision-making. These modules are interconnected through well-defined input–output interfaces, ensuring seamless data flow, iterative learning, and coordinated operation across the entire digital twin system.

Figure 3: Overall System-Level Control Optimization Architecture Flowchart. It shows the overall system architecture of the proposed work. Please click here to view a larger version of this figure.

The system architecture shows a complete picture of the model that can be used for modeling and optimizing complex nonlinear physical dynamics in oil and gas production systems. This starts with the input data, which includes the parameters used for the simulation and the boundary conditions used to represent the physical environment. This data first passes through the data preprocessing module, where normalization and feature extraction are done. This prepares the data for the next module, where the neural operator, FNO, and Neural operator model are used. These models are used to predict the system states, and this information is used to optimize the system parameters using the optimization module. The optimized results are then used to make decisions using the decision-making module. This module helps determine the efficiency of the system. The framework also includes the output results, where the optimization process results are shown in Figure 3. In addition, a feedback loop is included, where the data passes through the preprocessing module after the output results.

Results

In order to test the validity of the physics-informed neural operators, additional performance tests were conducted to compare the performance of FNO and Neural operator model in modeling the dynamics of the nonlinear multiphase flow system. The results showed that FNO performs better in global spatial consistency, while Neural operator model performs better in adaptability to heterogeneous inputs.

To guarantee effective training and assessment of the suggested digital twin framework, all experiments were carried out in a high-performance computing environment. Python was used as the main programming language for the implementation. A popular deep learning framework with GPU acceleration enabled by CUDA and an optimized deep neural network library was used to create deep learning models, such as physics-informed neural operators, spatiotemporal graph neural networks, and reinforcement learning modules. In order to facilitate large-scale multivariate time-series data processing and model training, the tests were carried out on a workstation with a multi-core CPU, a high-performance graphics processing unit (GPU) with dedicated memory, and at least 64 GB of system memory. To ensure compatibility with the deep learning ecosystem, the operating system environment was built on a Linux distribution.

Mini-batch gradient-based optimization was used to train the models, and numerous independent runs of the experiments were conducted to guarantee statistical reliability. The suggested methodology can be replicated in comparable high-performance computing settings thanks to the computational setup that has been provided.

To make the results validated through repeated trials, all experiments were repeated with multiple independent runs with different random initialization values. In the case of experiments using simulated data, the model has been executed for (N) independent trials, and the values are reported with the mean performance and standard deviation. In the case of the experiments using the benchmark data, SWAT and WADI, the evaluation metrics MAE, RMSE, F1 Score, and AUROC have been calculated on the test data and averaged over the repeated experiments to make the results statistically robust. This was done to obtain a reliable estimate of the performance of the model and to make sure that the performance improvements are not due to any single experiment.

Nonetheless, the outcomes have verified that merging physics-informed learning, spatiotemporal modeling, and data assimilation techniques in an integrated digital twin system consistently results in improved monitoring performance under the evaluated conditions, faster convergence, and optimal decisions in real time compared to conventional methods. Table 4 shows the simulation environment of proposed work.

The simulation environment is modelled to accurately resemble real-time operation within industry. Data from sensors is taken at short intervals to represent the stream data environment, and controlled amounts of noise are added to test robustness. Various fault and disturbance cases are simulated to test anomaly detection and control capabilities. A sliding window approach for evaluation is used to represent real-time operation in a continuous manner suitable for dynamic updates by the proposed digital twin for predictions and state synchronizations and for producing control actions.

In the design of the control-aware digital twin model uses several evaluation criteria must be considered for simultaneous assessment. First, for the monitoring and state prediction power of the digital twin architecture, regression criteria such as Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE) are used. MAE and MAPE calculate the accuracy of neural operators and spatio-temporal graph models on key system parameters like pressure, rate of flow, and temperature, and it is valid for the continuous multivariate data of sensors used in the oil and gas production process.

Secondly, the synchronization process of the digital twin itself, concerning state estimation error and prediction drift removal, is also assessed. The state estimation error measures the degree of error among the actual scheme state and the synchronized digital twin state, which is an indicator of how effectively the developed neural 4D-Var module uses the actual system state observations to improve the predictions of system dynamics. Finally, the prediction drift reduction indicates the gain in performance of the proposed approach against the unsynchronized digital twin system due to the ability of the approach to withstand noisy or sparse system observations.

Third, when it comes to detecting anomalies and fault diagnosis, classification-oriented evaluation parameters such as Precision, Recall, F1, and Area Under the ROC Curve (AUC) are employed. While recall refers to the ability to accurately diagnose real errors without overlooking any, precision can be seen as a measure of the validity of a recognized abnormality. The performance can be aptly gauged by using an F1 metric. The evaluation parameters are generally adopted in a SWAT and WADI system.

Table 5 is used to assess the efficacy of the proposed neural 4D-Var synchronization module for the reduction of prediction drift between the physical and cyber models for three datasets. The initial state error corresponds to the error between models without synchronization, and the final state error corresponds to the model error after applying synchronization. It can be seen that there is a significant improvement in state error for all three datasets, with the highest improvement (72.2%) occurring for the simulated oil and gas system model due to its relative simplicity compared to SWAT (63.6%) and WADI (57.7%). These readings are consistent with their respective levels of complexity and levels of “noise.” The increased rate of drift reduction for the simulated oil and gas dataset may be explained by the fact that the simulated environment generally involves a reduced level of noise, fewer random perturbations, and well-defined system behavior when compared to other realistic cyber-physical systems like SWAT and WADI. However, this argumentation should be treated as a hypothesis and not be quantitatively proven based on the results of this work. In future work, complexity measures like entropy, noise variance, and system dimensionality may be used to validate this argument.

Figure 4: Monitoring performance across different datasets (Simulated Oil and Gas, SWAT, and WADI) evaluated using error metrics MAE, RMSE, and MAPE (%). Bars represent mean values obtained from multiple independent runs (n = 5). Error bars indicate standard deviation (SD), while thinner overlaid error bars represent the standard error of the mean (SEM = SD/√n). The SD and SEM values for each metric are explicitly shown in the legend. Statistical significance is assessed using independent t-tests with respect to the baseline dataset (Simulated Oil and Gas), and corresponding p-values are displayed above the bars. Please click here to view a larger version of this figure.

The performance analysis of the monitoring process performed by the proposed digital twin model on the datasets such as Simulated Oil and Gas, SWAT, and WADI, taking into consideration the evaluation criteria of MAE, RMSE, and MAPE, has been represented in Figure 4. The simulated Oil and Gas dataset has the lowest error measure for all criteria, which clearly manifests the accuracy of system state monitoring in a simulated environment where the system parameters are modeled precisely. The SWAT and WADI datasets demonstrate errors in system monitoring that are gradually increasing due to the complexities of real systems together with their innate noise and nonlinearities, which are not found in simulated systems. However, the minor elevation in the error measures of MAE and RMSE confirms the stability of system performance of the digital twin model in such circumstances.

Figure 5: Anomaly detection performance across datasets (Simulated Oil and Gas, SWAT, and WADI) evaluated using F1-score and AUC. Bars represent mean values obtained from multiple independent runs (n = 5). Error bars indicate SD, while thinner overlaid error bars represent the SEM = SD/√n. Please click here to view a larger version of this figure.

Figure 5 shows the evaluation analysis of the suggested system for anomaly discovery based on F1-score and AUC for the three datasets. The Oil and Gas dataset, which is the simulated dataset, performs the best as it scores the highest F1-score and AUC. Although there is a gradual decline in performance, it achieves high AUC performance across datasets, with slight variations depending on dataset complexity. The alignment of both F1-score and AUC curves for the three datasets shows an optimal striking point, which reveals a good trade-off and makes the proposed system a reliable system for early fault detection and intrusion. The proposed system performs outstandingly in recognizing anomalies for the three datasets.

Precision, Recall, and F1-Scores of the proposed digital twin model on detecting anomalies on different datasets are shown in Table 6 below. Although high precision values show that most of the points of interest identified are indeed true points of interest, high values of recall show that most of the actual points of interest are correctly identified without missing out on important ones. The demonstration environment on the oil and gas domain registers the highest F1-score of 0.94, while slightly lower results on SWAT and WADI can be attributed to higher levels of noise and interdependencies on sensors and attack patterns. However, high F1-scores validate that robust spatio-temporal graph modeling and synchronization for detecting anomalies is attained.

Figure 6: Receiver Operating Characteristic (ROC) curves for anomaly detection across Simulated Oil and Gas, SWAT, and WADI datasets. The performance is evaluated using the Area Under the Curve (AUC). The standard deviation (SD) and standard error of the mean (SEM = SD/√n, n = 5) of AUC values are included in the legend for each dataset. The diagonal dashed line represents random classification performance. Please click here to view a larger version of this figure.

Figure 6 shows the Receiver Operating Characteristic (ROC) curves, along with the associated AUROC measures for the three different datasets, assessing the anomaly detection accuracy independently of the choice of threshold. In the ROC for the simulated Oil and Gas dataset, the region comes remarkably close to origin, thereby establishing an AUROC of 0.99, which confirms the virtually flawless classification capability. In the case of the SWAT and WADI datasets, the AUROC values lie at 0.97 and 0.95, respectively, thereby establishing good classification capability despite the presence of higher uncertainties. The ROC curves being far away from the lines representing random classification guarantee that the developed digital twin model performs remarkably well than the chance level for accurate anomaly detection. The Receiver Operating Characteristic (ROC) curves in Figure 6 show the performance of the proposed digital twin model in anomaly detection on the three datasets. The AUROC scores are 0.99 for the Oil and Gas simulated dataset, 0.97 for the SWAT dataset, and 0.95 for the WADI dataset, indicating the discrimination ability of the model in both simulated and benchmark industrial control systems.

Table 7 presents a concise description of the cumulative rewards and constraint violations for the reinforcement learning module for closed-loop control integrated with the digital twin. The cumulative reward obtained represents the overall control performance based on operational goals, while constraint violations represent the total number of violations during operation. The cumulative rewards with zero constraint violations are highest (96.5) for our oil and gas model environment, establishing optimal control and operational performance.

Even though the SWaT and WADI datasets show slightly lower cumulative rewards and some constraint violations, the overall improvement in operational cost savings demonstrates that the proposed digital twin framework remains effective for safe and optimal control in dynamic environments.

The real-time feasibility of the proposed reinforcement learning (RL)-driven control system was assessed by analyzing the end-to-end inference latency of all the computational modules used within the proposed system. The experimental observations in Table 8 show that the neural operator module, FNO/Neural operator model, and the spatio-temporal graph neural network (ST-GNN) module have moderate computational overheads due to the temporal features used during the process. On the other hand, the reinforcement learning policy decision module has a very light computational overhead. The cumulative end-to-end inference time of all the modules used within the proposed system is within the range of the typical industrial control cycles. Specifically, the average end-to-end inference latency is observed to be within 50–150 milliseconds per control cycle, which is compatible with standard industrial control requirements. This shows that the proposed system can be used for real-time and near real-time applications. However, note that the end-to-end inference time may vary based on the hardware used. In this regard, the proposed system can be used for real-time applications.

A quantitative comparison of the FNO, Neural operator model, and baseline approaches, such as traditional numerical solvers and current operator-learning techniques, is carried out to further establish the efficacy of physics-informed neural operators. The assessment from Table 9 focuses on how well each model captures the system's dynamic and nonlinear dynamics. Standard metrics including mean squared error (MSE), mean absolute error (MAE), and relative error between expected and ground-truth system states are used to evaluate performance. According to experimental results, both FNO and Neural operator model outperform baseline approaches in modeling complicated multiphase dynamics by achieving reduced error values. Specifically, Neural operator model successfully captures nonlinear operator mappings under different input conditions, while FNO performs well in learning global spatial dependencies.

The efficacy of the suggested neural operator models is shown by the quantitative comparative results shown in Table 9. When compared to baseline numerical solvers and current operator-learning techniques, both FNO and Neural operator model achieve noticeably lower error metrics. In particular, FNO has the lowest relative error and mean squared error, demonstrating its superior capacity to capture the dynamics of complicated nonlinear systems. Additionally, Neural operator model demonstrates significant gains, especially when it comes to learning nonlinear operator mappings under different circumstances. Independent t-tests are used to assess statistical significance; p-values less than 0.05 signify a substantial improvement over the baseline numerical solver. The p-values obtained by both suggested models are less than 0.001, indicating that the observed improvements in performance are statistically significant and not the result of chance. Furthermore, the suggested techniques show significantly shorter inference times, demonstrating their applicability to real-time digital twin applications.

To further improve the clarity of the experimental evaluation of the proposed control-aware digital twin model, a comprehensive quantitative comparison of the proposed approach with other methods is provided in an integrated tabular format. This comparison of various performance parameters such as Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), F1-score, Area Under the Curve (AUC), reduction of prediction drift, and cumulative reward will enable an integrated evaluation of monitoring accuracy, anomaly detection capability, synchronization efficiency, and control performance of the methods in Table 10. Additionally, various parameters of the training procedure such as batch size, learning rate, number of epochs, and module-wise training configuration are explicitly mentioned to improve the transparency of the experimental evaluation in Table 11. Further, scalability and computational efficiency in terms of inference latency, GPU utilization, and real-time feasibility of the proposed approach are also evaluated in Table 12.

The proposed digital twin can be comprehensively compared with other methods such as physics-based synchronization method, LSTM and GNN model in terms of monitoring accuracy, synchronization ability, anomaly detection, and control performance, as depicted in Table 13. Physics digital twins are less adaptive and have higher error margin based on static approaches. Neural approaches, such as LSTM and GNN, are better but not effectively combining physics, synchronization, and control. The proposed digital twin has achieved the best value in 0.20 of RMSE, 72.2% in drift reduction, 0.94 in AUROC, 0.92 in F1-Score, and 95.5 in cumulative reward. The above results clearly indicate that effectively combining physics-informed neural operators, graph modeling for space and time, neural data assimilation, and reinforcement learning can significantly improve an accurate, robust, and control-aware digital twin system.

Figure 7: Training cost and resource utilization of different components of the proposed system, including FNO, Neural operator model, GNN, RL module, and the total framework. Metrics include training time (hours) and GPU memory usage (GB). Values represent mean results over multiple independent runs (n = 5). Error bars indicate SD, while thinner overlaid error bars represent the SEM = SD/√n, reflecting variability across runs. Please click here to view a larger version of this figure.

In Figure 7, the training cost and resource utilization of the proposed approach are presented as their major components. It can be observed that the neural operator models, such as FNO and Neural operator model, have significant training costs and GPU memory utilization compared to the overall system. The significant training cost is due to the high-dimensional learning capability of the operators. The graph neural network and reinforcement learning modules are found to have lower training cost, but they are significant for the learning of spatial dependencies and adaptive control. Although the proposed system has higher training costs, the inference time is found to be efficient. Compared to baseline methods such as standalone LSTM or GNN models, the proposed system incurs higher training cost due to multi-module integration, but achieves improved accuracy and robustness, as demonstrated in Table 13.

Overall, the suggested digital twin structure combines several elements, each of which tackles a distinct problem in the modeling and control of intricate industrial systems. In contrast to traditional numerical solvers, which are frequently computationally costly and less appropriate for real-time applications, physics-informed neural operators, such as Fourier Neural Operator and Deep Operator Network, are used to capture nonlinear and high-dimensional system dynamics more effectively. Simpler machine learning models, such feedforward or recurrent neural networks, can be employed, but they usually don't learn fundamental physical operators and can't generalize across different system configurations. Dependencies between distributed sensors, which are intrinsically linked in oil and gas production systems, are modeled using the spatio-temporal graph neural network. Conventional methods, like independent time-series models or simple recurrent architectures, may lead to lower accuracy in state estimation and anomaly detection since they do not explicitly reflect spatial linkages. Adaptive control, which makes use of the reinforcement learning module, allows the system to maximize operational choices in dynamic and uncertain environments. Rule-based or static control strategies, on the other hand, are rigid and unable to adjust to changing system conditions or unanticipated disruptions. While there are alternatives with less complexity, they typically only deal with certain parts of the issue and are insufficient to achieve an integrated solution that incorporates precise modeling, real-time monitoring, and adaptive control. By ensuring that every part contributes to the overall system performance, the suggested modular design improves accuracy, scalability, and robustness in challenging industrial settings.

Study / Work	Techniques Used	Outcomes	Limitations
Deep Neural Operator for Digital Twin26	DeepONet neural operator as surrogate model	Real-time prediction; orders-of-magnitude faster than physics simulators	Needs careful sensor placement; evaluation methods need improvement
Virtual Sensing-enabled Digital Twin with DeepONet27	DeepONet virtual sensors in DT	Fast predictions; real-time inference; comprehensive state estimation	Spectral bias affects high-frequency phenomena; hybrid modeling needed
Digital Twin with DRL + NMPC in Control28	Deep Reinforcement Learning & Nonlinear MPC integration	Improved control performance; lower tracking error and adaptive twin updates	Application limited to vessels; further application to production systems needed
AI-Driven Digital Twin in Industrial Automation29	Supervised/Unsupervised ML (LSTM, CNN, RF, etc.)	High accuracy in fault prediction & monitoring	Primarily classic ML; lacks advanced spatio-temporal deep models
Oil & Gas Digital Twin Review16	Various AI/ML & mechanistic modeling	Comprehensive taxonomy of modeling strategies	Mostly conceptual; minimal practical implementations with advanced DL

Table 1: Comparative analysis of oil and gas production and control management with existing methods. An overview of relevant digital twin and control strategies, emphasizing methods employed, results attained, and significant drawbacks.

Dataset Name	Domain / System	No. of Features	Duration & Size	Data Characteristics
Simulated Oil & Gas Production Data	Oil & Gas Production System (Simulation)	Multivariate (Pressure, Flow, Temperature, etc.)	Designed for this research	Simulated real-time sensor & control signals, normal & fault states
SWaT	Secure Water Treatment CPS	~51 sensors + actuators	~11 days (~450k+ samples)	Normal & anomaly (36 attack scenarios)
WADI	Water Distribution CPS	~123 sensors + actuators	~16 days (~1M+ samples)	Normal & anomaly (15 attacks)

Table 2: Dataset description.
Dataset Description. Summary of the study's datasets, including domain, feature count, duration, and data attributes.

Component	Hyperparameter	Value
ST-GNN	Number of layers	3
ST-GNN	Hidden dimension	64
ST-GNN	Temporal window length	12
Neural Operator (FNO)	Number of Fourier modes	16
Neural Operator (FNO)	Number of layers	4
DeepONet	Branch network size	[128, 128]
DeepONet	Trunk network size	[128, 128]
Data Assimilation	Assimilation window	10 time steps
Optimizer	Adam
Learning rate	0.001
Batch size	64
RL Controller	Discount factor (γ)	0.99
RL Controller	Policy network layers	[128, 64]

Table 3: Model Architecture and Hyperparameters. Important architectural configurations and hyperparameter settings for the neural operators, data assimilation, reinforcement learning, and spatiotemporal graph neural network components.

Parameter	Description
Simulation platform	Python (v3.10) with PyTorch (v2.1)
Supporting libraries	NumPy (v1.24), SciPy (v1.10), Pandas (v1.5)
Hardware	NVIDIA RTX series GPU (e.g., RTX 3090, 24 GB VRAM), Intel Core i7/i9 CPU, 32–128 GB RAM
GPU acceleration	CUDA Toolkit (v11.8), cuDNN (v8.x)
Operating system	Linux (Ubuntu 20.04) / Windows 10
Development environment	Jupyter Notebook / VS Code
Sampling interval	1–10 seconds (configurable for real-time simulation)
Sensor noise modeling	Gaussian noise (σ = 0.01–0.05)
Fault scenarios	Sensor faults, flow disturbances, and control anomalies
Evaluation mode	Sliding window-based real-time simulation
Code availability	Repository link will be provided upon request / publication for reproducibility

Table 4: Simulation Environment. Information about the hardware configuration, software platform, sampling interval, noise modeling, and fault scenarios in the experimental setup.

Dataset	Initial State Error	Final State Error	Drift Reduction (%)
Simulated Oil & Gas	0.9	0.25	72.2
SWAT	1.1	0.4	63.6
WADI	1.3	0.55	57.7

Table 5: Experimental result of Digital Twin Synchronization. Quantitative assessment of synchronization performance that displays prediction drift reduction across datasets and initial and final state estimate errors.

Dataset	Precision	Recall	F1-Score
Simulated Oil & Gas	0.95	0.93	0.94 ± 0.01
SWAT	0.9	0.88	0.92 ± 0.02
WADI	0.87	0.85	0.90 ± 0.02

Table 6: Anomaly detection performance. Results for anomaly detection on simulated oil and gas, SWAT, and WADI datasets, including precision, recall, and F1-score.

Dataset	Cumulative Reward ↑	Constraint Violations ↓	Cost Reduction (%) ↑
Simulated Oil & Gas	96.5 ± 0.8	0	18.7
SWAT	92.1 ± 1.2	1	14.3
WADI	89.7 ± 1.5	2	11.6

Table 7: Experimental results of Closed Loop control and Optimization. Cumulative reward, constraint violations, and operational cost reduction are used to evaluate closed-loop control performance.

Module	Average Latency (ms)	Description
Neural Operator (FNO/DeepONet)	18-25 ms	Operator learning and system state prediction
ST-GNN	22–35 ms	Spatio-temporal dependency modeling
RL Policy Network	5–10 ms	Control decision generation
Data Preprocessing	8–12 ms	Input normalization and feature preparation
Total Control Loop Latency	55–82 ms	End-to-end control action execution time

Table 8: Real-time control performance evaluation. Latency values represent average inference times measured per control step under a standard hardware configuration.

Model	MSE ↓	MAE ↓	Relative Error (%) ↓	Inference Time (ms) ↓	p-value (vs Baseline)
Numerical Solver (Baseline)	0.0125 ± 0.0012	0.089 ± 0.006	8.75 ± 0.54	120.5 ± 5.2	—
Conventional ML Model	0.0098 ± 0.0009	0.072 ± 0.005	6.42 ± 0.48	85.3 ± 4.7	0.021
Existing Operator Learning	0.0076 ± 0.0007	0.061 ± 0.004	5.18 ± 0.41	42.8 ± 3.1	0.008
DeepONet (Proposed)	0.0052 ± 0.0005	0.044 ± 0.003	3.67 ± 0.32	18.6 ± 2.4	< 0.001
FNO (Proposed)	0.0047 ± 0.0004	0.039 ± 0.002	3.21 ± 0.28	15.2 ± 2.1	< 0.001

Table 9: Comparison of baseline and proposed models in terms of prediction accuracy (MSE, MAE, Relative Error) and computational efficiency (Inference Time). Values are reported as mean ± standard deviation over multiple runs. Lower values indicate better performance. p-values denote statistical significance compared to the baseline numerical solver.

Method	MAE ↓	RMSE ↓	F1-Score ↑	AUC ↑	Drift Reduction (%) ↑	Cumulative Reward ↑
Physics-based DT	0.45	0.62	0.78	0.82	35.4	65.2
LSTM	0.32	0.48	0.85	0.88	48.7	74.5
GNN	0.28	0.41	0.88	0.91	55.3	81.3
Proposed Framework	0.2	0.3	0.94	0.97	72.2	95.5

Table 10: Comprehensive performance comparison. Comprehensive performance comparison of the proposed control-aware digital twin model with other baseline approaches for monitoring, anomaly detection, synchronization, and control. For prediction accuracy, lower values of MAE and RMSE are preferred. For anomaly detection and synchronization performance, higher values of F1-score, AUC, drift reduction, and reward are preferred.

Component	Parameter	Value/Setting
Neural Operators (FNO/DeepONet)	Epochs	100–200
	Batch Size	32–64
	Learning Rate	0.001
ST-GNN	Layers	3
	Hidden Units	64
	Temporal Window	10–20 timesteps
Neural 4D-Var	Assimilation Window	Fixed (sliding window)
	Optimization	Gradient Descent
Reinforcement Learning	Discount Factor (γ)	0.95
	Reward Weights (α,β,γ)	0.6, 0.25, 0.15
General Training	Optimizer	Adam
	Hardware	NVIDIA RTX GPU

Table 11: Training configuration and implementation parameters used for different components of the proposed digital twin framework. The table provides an overview of different hyperparameters and training parameters used to ensure reproducibility of the proposed system for different components.

Component	Training Time (hrs)	Inference Latency (ms)	GPU Utilization (%)	Scalability
Neural Operators	6–8	25–40	80–90	High
ST-GNN	3–4	15–25	60–70	High
Neural 4D-Var	2–3	20–30	65–75	Moderate
RL Controller	1–2	10–20	50–60	High
Overall System	12–16	<100 ms	70–85	Scalable

Table 12: Computational performance and scalability analysis of the proposed digital twin model for different components. Latency is used to represent real-time inference performance, GPU is used to represent computational performance, and scalability represents the proposed model’s ability to be used for industrial-scale applications.

Method	Monitoring RMSE	Synchronization Drift Reduction (%)	AUROC	F1-Score	Cumulative Reward
Physics-based Digital Twin	0.45	28.6	0.78	0.74	42.3
LSTM-based Model	0.31	41.2	0.86	0.83	61.7
GNN-based Model	0.26	53.4	0.91	0.88	74.5
Proposed Digital Twin	0.2	72.2	0.94	0.92	95.5

Table 13: Comparative evaluation of proposed work with existing methods. The suggested digital twin model's performance in terms of monitoring, synchronization, anomaly detection, and control metrics is compared against baseline and learning-based approaches.

Model Configuration	Monitoring Error (RMSE)	Anomaly Detection (F1)	Control Performance (Reward)
Full Model (All Modules)	0.2	0.92	95.5
– Without Data Assimilation	0.29	0.88	89.2
– Without ST-GNN	0.27	0.85	87.1
– Without Reinforcement Learning	0.21	0.9	78.4
– Without Continuous Learning	0.23	0.91	92.3

Table 14: Experimental results and Ablation Study. Ablation study examining how each part of the suggested digital twin model contributes to the overall performance of the system.

Scenario	Data Availability	RMSE	Accuracy	Observation Gap Handling
Normal	100%	Low	High	Stable
Partial Loss	50–70%	Moderate	High	Robust interpolation
Severe Loss	20–30%	Increased	Moderate	Graceful degradation
Sensor Failure	Region-wise 0%	Higher	Moderate	Spatial generalization

Table 15: Robustness analysis under data scarcity. The following table indicates the performance of the proposed neural 4D-Var framework in different situations of data availability. The table indicates the robustness of the model by showing how errors change in situations of partial loss, high sparsity, and total loss of sensor data.

Discussion

The findings from this research demonstrate the effectiveness of the developed control-aware digital twin in terms of its ability to improve real-time monitoring, detection, and control optimization for a wide range of CPS. Compared to the traditional models for monitoring, the combination of the PINO and the graph neural networks is more effective in terms of accuracy for state estimation, even for a noisy system. This finding is true because an advanced machine learning model in the digital twin is more effective in terms of prediction and detection in the system³⁹.

Compared to current solutions for anomaly detection, such as digital twin anomaly detectors employing curriculum learning or causal representation models, it can be noticed that the proposed method can achieve competitive performance by comparing the results obtained in literature. For example, digital twin-based anomaly detectors employing curriculum learning enabled a moderate enhancement of F1-score for CPS testbeds, which indicates that learning paradigms can affect the efficiency of detection processes⁴⁰. At the same time, the current popular digital twin models strongly emphasize efficient reasoning and the suppression of false positives via causal inference, but at the cost of bigger model difficulty in the causal model. In this regard, our proposed synchronized reinforcement learning approach supplies both efficient operation optimization and sound anomaly detection within a solitary background. It should be noted, however, that the comparison of the proposed method with other approaches is not based on the results of direct comparative experiments, but only on the results reported by other researchers; therefore, the comparison should be made with this consideration.

Moreover, the addition of anomaly detection and optimization of control actions within the proposed digital twin illustrates the growing importance of decision-support and actuation capabilities provided by digital twins, a current direction of research and development activities associated with CPS. Research papers⁴¹ related to digital twin calibration using reinforcement learning, such as adaptive approaches for guiding sensors, show that the use of learning agents can provide added benefits, especially regarding accuracy and adaptability, especially for dynamic systems. This confirms the findings regarding the enhancement of the effectiveness of closed-loop optimization, thus reinforcing the hypothesis that intelligent integration can increase CPS robustness and efficiency. The approach, as mentioned earlier, is designed to work for the oil and gas industry, but the evaluation of the approach is based on the simulation of the oil and gas industry, as well as the benchmark datasets of CPS, i.e., SWaT and WADI. However, it should be noted that the current validation does not include real-world oil and gas operational data, and therefore the applicability to real deployment scenarios should be interpreted cautiously. In the future, comparative evaluation of the proposed curriculum learning method and the digital twin method based on causal inference will be made under a unified experimental framework to compare the two approaches in a controlled manner. Although the integration of these modules, including neural operators, graph networks, and reinforcement learning, increases the computational cost, each module has a specific function to perform. The ablation study also validates the performance of the proposed approach, as the performance degrades if any of the modules are removed.

Theoretically, this research makes significant contributions to the addition of model-based and data-driven methods within the digital twin structure. The integration of physics-informed neural operators and neural data assimilation techniques into the state estimation process provides a hybrid modeling approach for CPS, thus filling a significant gap in the digital twin research literature, which has relied either on purely physics-based simulations or black-box models for CPS simulations but not both. Theoretically, this research aligns with the progression of digital twin research for autonomous CPS models that can continuously learn and adapt, which has been emphasized in recent literature.

In terms of applicability, the proposed approach can greatly improve the working of CPS in the industrial setting as it enables real-time reliability, rapid and precise anomaly identification, and optimal control loop strategies as a result of direct augmentation of safety and efficiency in the operation of CPS. For example, in the industrial setting where the environment is likely accompanied by noise in sensors and a nonlinear system model, the synchronized detection approach ensures rapid identification of abnormalities while suppressing the occurrence of false alarms. As opposed to existing digital twin-based CPS systems’ solutions on intrusion detection, aimed chiefly at classification metrics, the proposed approach enables both identification and correction approaches, contributing to the ability of the system’s operators to proactively alter the operating environment of the CPS before the occurrence of downtime. In addition, the capability of the suggested approach to universally apply in dissimilar settings as identified by its performance on the benchmark dataset indicates the overall applicability of the approach in different CPS settings like smart manufacturing, the energy domain, and critical infrastructure network management systems.

The suggested digital twin framework's scalability is assessed in terms of its capacity to manage growing system size and temporal complexity. By representing sensor interactions using sparse graph structures, the spatio-temporal graph neural network efficiently scales as the number of sensors and monitored components increases, preventing exponential growth in computing complexity. By learning operator mappings irrespective of discretization resolution, the application of physics-informed neural operators further improves scalability and enables effective modeling of large-scale systems without appreciable increases in computational cost. The framework uses sliding window methods and sequence-based learning to manage temporal dependencies over larger time horizons without sacrificing computing viability. Additionally, incremental learning techniques reduce overhead in large-scale deployments by enabling continual updates without the need for complete retraining. Furthermore, effective real-time processing of high-dimensional data streams is made possible by the utilization of parallel GPU-based computation. The suggested architecture is appropriate for large-scale industrial applications like oil and gas production systems with distributed sensor networks and continuous monitoring requirements since it is made to scale with both the number of sensors and the time of operation.

The ablation study will help in understanding the individual contribution of each module, i.e., neural operators, ST-GNN, data assimilation, and reinforcement learning. An ablation study has been carried out to measure the contribution levels of numerous individual major modules in this proposed digital twin architecture. These individual modules include data assimilation, graph modeling with space and time, reinforcement learning, and continuous learning. The effects of removing these individual modules have been examined in terms of effect levels they create in monitoring accuracy, anomaly detection, and control. The final results clearly validate that it is always better to make use of the overall architecture, and thus removing individual modules results in varying levels of performance degradation, indicating that different components contribute differently to the overall system performance. The removal effect level of the data assimilation module clearly shows an increase in prediction error and affects reliability at an equal level concerning anomaly detection. The effect level of removing reinforcement learning clearly shows that control has got significantly affected, especially when only monitoring is considered.

The ablation results show in Table 14 that different modules have different contributions to the performance of the system. The data assimilation module has the largest contribution to the accuracy of the monitoring performance, followed by the ST-GNN module. The continuous learning module has a moderate contribution to the accuracy of monitoring performance, and the reinforcement learning module has the smallest contribution to the accuracy of the monitoring performance, although it has the largest contribution to the control performance in terms of the drop in the reward value from 95.5 to 78.4.

To validate the robustness of the proposed neural 4D-Var framework, experiments were performed under increasingly deteriorated observation conditions. Specifically, the experiments included the following scenarios: (i) partial sensor failure, where the percentage of missing data ranged from 30% to 70%; (ii) full sensor dropout in specific regions; and (iii) communication delay resulting in temporally sparse observations. From the experimental results, the proposed model exhibits robust performance under moderate data loss and deteriorates gracefully under extreme data loss, illustrating its high level of adaptability. In Table 15 shows the result of Robustness Analysis.

A limitation of this study is associated with the scope of the validation of the proposed model. It should be noted that while the model is intended for oil and gas production systems, the evaluation of the proposed model was based on simulated oil and gas values and cyber-physical benchmark problems like SWaT and WADI. It is also important to note that while the benchmark problems are associated with oil and gas processes, in this case, the problems are associated with water treatment and distribution.

However, despite this optimal performance, it has some limitations. For instance, it has higher computational complexity compared to some digital twin implementations, especially with the integration of neural operators, graph networks, and reinforcement learning. This could be a major impediment in their adoption of platforms with limited computing capabilities. Moreover, although it has only focused on simulation and benchmark sets, they could not include the level of diversity that would be found in an operational CPS. This could be seen in simulation sets, such as SWAT and WADI, which are generally adopted in CPS assessments. They could lack the level of diversity that would be experienced in real-world CPS environments. It also assumes an appropriate supply of sensor data, which could not always be feasible in older CPS. The major limitation of the system is the lack of real-world oil and gas industry data to evaluate the system, as the evaluation of the system is based on the simulation of the oil and gas industry, as well as the benchmark datasets of CPS, i.e., SWaT and WADI, which might not accurately reflect the real-world complexities of the oil and gas industry. The proposed system has demonstrated promising performance in predictive capabilities. However, the proposed system requires higher computational costs during the training process. In the future, lightweight alternatives to the proposed system, including reduced-order neural operators, model pruning, and hybrid physics-ML models, can be considered.

Scalability and deployment in a real-world environment, and its integration in existing legacy CPS infrastructures where the sensor density and quality of the sensed information can vary greatly, need to be studied in future research works on model compression techniques. Moreover, extension of this system to include explainability and feedback loops for operators could be beneficial in improving explainability and building trust in autonomous decision-making. In addition, integration of causal inference methods within the current anomaly detection processes, similar to those in causal digital twins, could be beneficial in improving root cause analysis and minimizing false alarms. Finally, in order to create robustness, this framework will need to be validated using actual field data.

This research has demonstrated the idea of an intelligent, control-aware digital twin model that aims to improve real-time system monitoring, anomaly detection, and decision-making capabilities in complex cyber physical systems. In contrast to traditional digital twins that are mainly passive observers and are used for real-time monitoring, this research aims to integrate physics-informed neural operators, spatio-temporal graph neural networks, neural data assimilation, reinforcement learning-based control, and learning algorithms within a single model of a digital twin system. The system is first monitored in real-time on a physics-informed learning model of a nonlinear system of equations by a digital twin system. Second, a neural 4DVar system is applied for a data assimilation procedure that synchronizes both physical and virtual systems to minimize the drift of system predictions that result from noise and system change. Anomaly detection is carried out on a spatio-temporal model of either spatial relations or system evolution over time of a system. Finally, a reinforcement learning agent creates system control actions from a synchronized system state that is valid from a system safety constraint perspective while a control constraint is considered. Extensive experiments conducted on a simulated oil and gas process, and the benchmark CPS scenarios SWAT and WADI, demonstrated the potential effectiveness of the proposed approach under simulated and benchmark conditions. The proposed model resulted in reduced monitoring errors (MAE and RMSE), excellent anomaly detection results and consistent behavior for closed-loop control. These findings have confirmed that the synchronization of anomaly detection and control within the digital twin is an important factor in relation to maintaining its reliability and efficiency of operations. Another area of potential development of digital twin technology is its extension by artificial intelligence and causality. The evaluation of the framework will be done by validating the framework using real-world data of the oil and gas industry.

It is important to emphasize that the conclusions drawn in this study are based on simulated and benchmark datasets, and therefore the applicability to real-world oil and gas systems should be interpreted with caution.

DATA AVAILABILITY:

The SWaT and WADI datasets used in this study are publicly available industrial control system datasets and can be accessed via the iTrust Lab dataset repository: https://itrust.sutd.edu.sg/itrust-labs_datasets/dataset_info/.

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
CPU (Intel Xeon / AMD Ryzen)	Intel / AMD	Xeon / Ryzen Series	Used for system simulation, data preprocessing, and general computation
Deep Learning Framework (PyTorch 2.0)	Meta AI (PyTorch)	https://pytorch.org RRID:SCR_018536	Used for implementing neural networks, optimization, and training pipelines
Deep Neural Network Library (cuDNN 8.x)	NVIDIA cuDNN	cuDNN 8.x	Provides optimized primitives for deep learning operations on GPU
DeepONet	Not Applicable	https://arxiv.org/abs/1910.03193	In the script, it is refered as "neural operator model"
GPU (NVIDIA RTX 4090)	NVIDIA	RTX 4090	Used for training deep learning models including neural operators, ST-GNN, and reinforcement learning components
GPU Acceleration (CUDA 11.8)	NVIDIA CUDA	CUDA 11.8 RRID:SCR_018131	Enables GPU-based parallel computation for high-performance model training
Linux distribution (version 22.04 LTS)	Canonical (Ubuntu)	https://ubuntu.com/download; RRID:SCR_018127	Operating system used for executing experiments and model deployment
Operating System (Ubuntu 22.04 LTS)	Canonical (Ubuntu)	https://ubuntu.com/download RRID:SCR_018127	Provides execution environment for all experiments and model deployment
Optimization Algorithm (Adam Optimizer)	Not Applicable	https://arxiv.org/abs/1412.6980	Used for training models with adaptive learning rate optimization
optimized deep neural network library (version 8.x)	NVIDIA cuDNN	https://developer.nvidia.com/cudnn RRID:SCR_018131	Provides optimized GPU-accelerated primitives for deep neural network operations
Programming Language (Python 3.10)	Python Software Foundation	https://www.python.org/downloads RRID:SCR_008394	Core programming language used for implementation and experimentation
System Memory (≥ 64 GB RAM)	Corsair / Kingston / Crucial	https://www.corsair.com / https://www.kingston.com / https://www.crucial.com	Supports handling of high-dimensional multivariate time-series data and large model training