## 1 Introduction  
As industrial systems become increasingly complex, traditional mechanistic modeling methods are struggling to address the significant challenges posed by modern production processes. This is especially true for systems characterized by strong multivariable coupling, nonlinearity, and time-variance, where the limitations of expert-based and traditional modeling methods have become increasingly evident. As a result, data-driven modeling approaches have gradually become the focus of research. By mining the vast amounts of data embedded in industrial production processes and combining data-driven models with optimization algorithms, the real-time monitoring, state prediction, control, optimization scheduling, and fault diagnosis of industrial processes can be achieved. This provides solid technical support for the intelligent transformation of modern manufacturing, while ensuring the stability of production processes and product quality [1, 2, 3].  
Traditional data-driven modeling and optimization algorithms primarily include multivariate statistical methods and machine learning models. Multivariate statistical methods, such as the use of one-way ANOVA for the detection of abnormalities in the sintering process [4] and Gaussian process regression for performance prediction in sintering [5], establish statistically based models to provide reliable decision support for industrial operations. With the rapid development of technologies such as big data, cloud computing, and the Internet of Things, researchers have become increasingly interested in the application of machine learning models in industrial production. Examples include resource allocation decision support systems that combine multi-objective optimization and meta-learning [6], artificial neural network optimization for solar energy multi-supply systems [7], and dynamic compressive strength prediction models for freeze-thaw rock based on a swarm intelligence optimization of hybrid support vector regression [8]. Machine learning has significant advantages in handling complex, dynamically changing data by automatically learning and adapting to different data patterns.  
As modern industry continues to advance toward automation and intelligence, advanced artificial intelligence techniques, such as deep learning, reinforcement learning, and ensemble learning, have become mainstream in data-driven intelligent modeling. Examples include the development of optimal control strategies for nonlinear industrial production systems using spatiotemporal graph convolutional networks [9] and reinforcement learning [10], as well as bearing fault detection methods using convolutional neural networks [11] and sintering endpoint prediction models [12]. Artificial intelligence methods are highly effective in processing high-dimensional, nonlinear, temporal, and dynamically changing data. They possess strong feature extraction and learning capabilities, enabling them to adapt to real-time and complex optimization problems in big data environments.  
Moreover, hybrid methods combining physical models with data-driven models can effectively integrate the advantages of physical models in causal analysis with the efficiency of data-driven methods in correlation analysis. For example, seismic damage prediction methods using finite element calculations in conjunction with multi-particle swarm optimization algorithms [13], integrated machine learning methods combined with physics-based empirical models for ship operation status recognition [14], and energy control strategies that combine mechanistic modeling with machine learning [15] all demonstrate the unique advantages of hybrid physical and data-driven models in addressing practical problems.  
However, numerous challenges have yet to be addressed, particularly when tackling high-dimensional, nonlinear, and multi-scale industrial problems. Therefore, further research efforts are essential to advance both the theoretical and practical developments in this critical field.