Machine learning (ML) provides powerful tools for industrial process modeling by identifying patterns and optimizing performance. However, conventional ML models often struggle with accuracy and reliability because they rely solely on data and fail to incorporate process-specific knowledge. To address this limitation, we propose a physics-informed ML approach that integrates engineering insights into an ML model. Specifically, we develop a variational autoencoder-based generative regression model with a probabilistic regression layer, enabling uncertainty-aware predictions. Additionally, we derive process knowledge through finite element analysis and incorporate it into the model via a custom physics-informed loss function, ensuring consistency with real-world process dynamics. The effectiveness of this approach is demonstrated in the steel strip rolling process, a critical manufacturing operation where precise flatness control directly impacts product quality. By integrating data-driven learning with physics-based constraints, the proposed method enables more accurate flatness predictions, reducing defects and improving process stability. This research provides a practical and scalable solution for industrial applications, offering manufacturers a more reliable tool for optimizing production processes and ensuring product quality.