###### Abstract
PINNs embed physical laws, such as differential equations, into the network architecture, improving model robustness and interpretability [11]. This hybrid approach bridges the gap between data-driven methods and physical system understanding, enhancing predictive accuracy for SAG mill operations [14; 15]. Furthermore, advanced class imbalance handling ensures sensitivity to rare overload events [10]. Model Architecture and Methodology: Our research is centered around implementing a convolutional neural network architecture, a powerful tool uniquely tailored for predicting SAG mill overloads. This architecture takes a novel approach by integrating physical rules directly into the learning process through the loss function. These rules, derived from the operational knowledge of SAG mills, are encoded within the model to guide the learning algorithm. This practical application of our research enhances the model's predictive accuracy and relevance to real-world operations, underlining the importance of our work to our audience in industrial process optimization and machine learning. The following are contributions of this work:  
* Integration of Physics-Informed Neural Networks (PINNs): We have developed a new way of using physics-informed neural networks (PINNs) in our CNN framework. Our method involves using physical principles to guide the learning process in solving both forward and inverse problems that involve differential equations. These principles help the model focus on important operational features that indicate overload conditions, such as the relationships between the mill's features. Through further investigation, we will determine if this technique continues to produce competitive results when compared to modifying the loss function, specifically ablating the physical principles from contributing to the model. * Practical Implications and Operational Insights: The paper discusses the practical implications of our research, emphasizing how the predictive model can be integrated into existing control systems to provide early warnings of potential overloads over features related to this type of event. * Interpretability: Our research pays special attention to establishing how the model is set to determine an overload. For this, we use modern interpretation techniques based on GradCam++ and thus identify what feature and at what moment it becomes relevant for the prediction, with appropriate changes to be adapted to this new proposal. * Contribution to the Field: This research contributes to the field by bridging the gap between advanced machine learning techniques and the practical and operational needs of the mining industry in the SAG ambit. It extends CNNs' application beyond traditional image and video processing to address complex industrial processes such as SAG mill operations. This proactive approach has the potential to significantly reduce downtime, optimize energy consumption, and extend the lifespan of mill components. In our research, we will address the importance of SAG mill overload forecasting as a critical factor in ensuring the efficiency of mill operations. Considering that existing approaches have made progress in addressing this issue of overload classification (actual time), our proposed method represents a significant advancement by integrating domain-specific physical knowledge to obtain predictions (future time). This integration will enhance the predictive accuracy and provide a more robust, reliable, and interpretable tool for mill operators and expert systems, contributing to more sustainable and efficient mining practices [16; 17]. The remainder of this paper is structured as follows: Section 2 discusses the problem of overload in SAG mills in more detail. Section 3 explore the basis of the physics informed neural networks (PINNs). Section 4 reviews related work on SAG mill overload prediction and the use of machine learning in this domain. Section 5 outlines the proposed method in detail, including the feature selection and encoding techniques.