π-Net: A Physics-Informed Machine Learning Architecture Integrating Dimensional Analysis and Directed Graphs for Interpretable Manufacturing Process Modelling

Integrating prior knowledge into machine learning for complex multi-physics manufacturing processes remains a fundamental challenge. Physics-informed machine learning (PIML) approaches partially address this challenge, yet most still require complete governing equations, limiting their applicability when only partial physical knowledge is available. This study proposes π-Net, a novel PIML architecture that integrates a machine learning backbone with a directed graph developed using the Dimensional Analysis and Conceptual Modelling (DACM) framework, enabling process-level explainability without requiring complete governing equations. The directed graph encodes physically directed dependencies derived from dimensional analysis, while the backbone learns unknown components within the directed structure directly from data, producing a fully self-contained system without reliance on simplifying assumptions. The framework is applicable to manufacturing processes for which a DACM directed graph can be constructed and is demonstrated on a Gas Metal Arc Welding (GMAW) case study for weld bead geometry prediction. Three backbone architectures are evaluated, demonstrating that the directed structure is the primary factor governing predictive performance. Based on 5-fold cross-validation over 60 experimental samples, the best-performing variant achieves an aggregate R² of 0.864 and MAPE of 6.93%, outperforming its purely data-driven counterpart across all training set sizes, with the advantage more pronounced under limited data conditions. The directed structure enables process engineers to trace how changes in process parameters propagate through the graph to influence process outputs. These results demonstrate that π-Net offers an effective approach for hybrid modelling where governing equations are incomplete and data is limited.