###### Abstract  
The overload problem in semi-autogenous grinding (SAG) mills is critical in the mining industry, impacting the extraction of valuable metals and overall productivity. Overloads can lead to severe operational issues, including increased wear, reduced grinding efficiency, and unscheduled shutdowns, which result in financial losses. Various strategies have been employed to address SAG mill overload, from real-time monitoring to predictive modeling and machine learning techniques. However, existing methods often lack the integration of domain-specific knowledge, particularly in handling class imbalance within operational data, leading to limitations in predictive accuracy. This paper presents a novel approach that integrates convolutional neural networks (CNNs) with physics-informed neural networks (PINNs), embedding physical laws directly into the model's loss function. This hybrid methodology captures the complex interactions and nonlinearities inherent in SAG mill operations and leverages domain expertise to enforce physical consistency, ensuring more robust predictions. Incorporating physics-based constraints allows the model to remain sensitive to critical overload conditions while addressing the challenge of imbalanced data. Our method demonstrates a significant enhancement in prediction accuracy through extensive experiments on real-world SAG mill operational data, achieving an F1-score of 94.5%. The results confirm the importance of integrating physics-based knowledge into machine learning models, improving predictive performance, and offering a more interpretable and reliable tool for mill operators. This work sets a new benchmark in the predictive modeling of SAG mill overloads, paving the way for more advanced, physically informed predictive maintenance strategies in the mining industry. machine learning; semi-autogenous grinding (SAG) mills; convolutional neural networks (CNNs); physics-informed neural networks (PINNs); predictive maintenance; overload prediction; mining industry; multi-channel CNN +
Footnote †: journal: applied sciences  
1
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
202411686  
11 December 20241 December  
role in ore processing for copper extraction. Their performance directly impacts mining productivity and profitability. With rising global copper demand driven by electric vehicles and renewable energy, SAG mill efficiency is increasingly crucial. A 1% improvement in treatment can yield up to USD 160 million in annual profits and cut CO\({}_{2}\) emissions by 200-400 tons without increasing electricity use [5; 6; 7]. However, operational challenges, particularly overloads, lead to costly disruptions, including unscheduled shutdowns and excessive component wear. These issues escalate energy consumption and operational costs, causing significant production losses. Historically, SAG mill overloads have been addressed using real-time monitoring systems that classify events as they occur [8]. Although useful, these methods lack proactive capabilities. Predictive modeling with machine learning, such as model predictive control (MPC) and advanced neural networks, offers improved forecasting [9; 10]. Yet, challenges persist due to operational variability and the nonlinear dynamics of SAG mill operations. Existing models often lack domain-specific knowledge and physical constraints, crucial for accurate overload prediction [11]. Additionally, class imbalance in data--where overload events are rare--biases models towards normal operations, reducing critical event detection performance [12; 13]. To address these gaps, we propose a method combining convolutional neural networks (CNNs) with physics-informed principles to enhance predictive accuracy. By integrating physical rules derived from domain expertise into the CNN loss function, we capture the nonlinearities of SAG mill processes through a physics-informed neural networks (PINNs) approach.