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Mathematics
Volume 13
Issue 15
10.3390/math13152411
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Review

From Block-Oriented Models to the Koopman Operator: A Comprehensive Review on Data-Driven Chemical Reactor Modeling

by
Mustapha Kamel Khaldi
1,2,
Mujahed Al-Dhaifallah
1,3,*,
Ibrahim Aljamaan
4,
Fouad Mohammad Al-Sunni
1,
Othman Taha
5 and
Abdullah Alharbi
6
1
Control and Instrumentation Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2
Interdisciplinary Research Center for Smart Mobility and Logistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3
Interdisciplinary Research Center for Sustainable Energy Systems, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4
Biomedical Engineering Department, Imam Abdulrahman Bin Faisal University, Dammam 31451, Saudi Arabia
5
Process & Control Systems Department, Saudi Aramco, Dhahran 31261, Saudi Arabia
6
Department of Accounting & Finance, KFUPM Business School, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(15), 2411; https://doi.org/10.3390/math13152411 (registering DOI)
Submission received: 10 June 2025 / Revised: 21 July 2025 / Accepted: 24 July 2025 / Published: 26 July 2025
(This article belongs to the Special Issue Recent Advances in Computational Intelligence Methodologies for Industries)
Versions Notes

Abstract

Some chemical reactors exhibit coupled dynamics with multiple equilibrium points and strong nonlinearities. The accurate modeling of these dynamics is crucial to optimal control and increasing the reactor’s economic performance. While neural networks can effectively handle complex nonlinearities, they sacrifice interpretability. Alternatively, block-oriented Hammerstein–Wiener models and Koopman operator-based linear predictors combine nonlinear representation with linear dynamics, offering a gray-box identification approach. This paper comprehensively reviews recent advancements in both the Hammerstein–Wiener and Koopman operator methods and benchmarks their accuracy against neural network-based approaches to modeling a large-scale industrial Fluid Catalytic Cracking fractionator. Furthermore, Monte Carlo simulations are employed to validate performance under varying signal-to-noise ratios. The results demonstrate that the Koopman bilinear model significantly outperforms the other methods in terms of accuracy and robustness.
Keywords: Deep Neural Network; Fluid Catalytic Cracking; Hammerstein–Wiener; Koopman operator; Long Short-Term Memory networks; modeling; review Deep Neural Network; Fluid Catalytic Cracking; Hammerstein–Wiener; Koopman operator; Long Short-Term Memory networks; modeling; review

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MDPI and ACS Style

Khaldi, M.K.; Al-Dhaifallah, M.; Aljamaan, I.; Al-Sunni, F.M.; Taha, O.; Alharbi, A. From Block-Oriented Models to the Koopman Operator: A Comprehensive Review on Data-Driven Chemical Reactor Modeling. Mathematics 2025, 13, 2411. https://doi.org/10.3390/math13152411

AMA Style

Khaldi MK, Al-Dhaifallah M, Aljamaan I, Al-Sunni FM, Taha O, Alharbi A. From Block-Oriented Models to the Koopman Operator: A Comprehensive Review on Data-Driven Chemical Reactor Modeling. Mathematics. 2025; 13(15):2411. https://doi.org/10.3390/math13152411

Chicago/Turabian Style

Khaldi, Mustapha Kamel, Mujahed Al-Dhaifallah, Ibrahim Aljamaan, Fouad Mohammad Al-Sunni, Othman Taha, and Abdullah Alharbi. 2025. "From Block-Oriented Models to the Koopman Operator: A Comprehensive Review on Data-Driven Chemical Reactor Modeling" Mathematics 13, no. 15: 2411. https://doi.org/10.3390/math13152411

APA Style

Khaldi, M. K., Al-Dhaifallah, M., Aljamaan, I., Al-Sunni, F. M., Taha, O., & Alharbi, A. (2025). From Block-Oriented Models to the Koopman Operator: A Comprehensive Review on Data-Driven Chemical Reactor Modeling. Mathematics, 13(15), 2411. https://doi.org/10.3390/math13152411

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