Mathematical Knowledge Graph-Driven Framework for Equation-Based Predictive and Reliable Additive Manufacturing

Additive manufacturing (AM) relies critically on understanding and extrapolating process-property relationships; however, existing data-driven approaches remain limited by fragmented knowledge representations and unreliable extrapolation under sparse data conditions. In this study, we propose an ontology-guided, equation-centric framework that tightly integrates large language models (LLMs) with an additive manufacturing mathematical knowledge graph (AM-MKG) to enable reliable knowledge extraction and principled extrapolative modeling. By explicitly encoding equations, variables, assumptions, and their semantic relationships within a formal ontology, unstructured literature is transformed into machine-interpretable representations that support structured querying and reasoning. LLM-based equation generation is further conditioned on MKG-derived subgraphs, enforcing physically meaningful functional forms and mitigating non-physical or unstable extrapolation trends. To assess reliability beyond conventional predictive uncertainty, a confidence-aware extrapolation assessment is introduced, integrating extrapolation distance, statistical stability, and knowledge-graph-based physical consistency into a unified confidence score. Results demonstrate that ontology-guided extraction significantly improves the structural coherence and quantitative reliability of extracted knowledge, while subgraph-conditioned equation generation yields stable and physically consistent extrapolations compared to unguided LLM outputs. Overall, this work establishes a unified pipeline for ontology-driven knowledge representation, equation-centered reasoning, and confidence-based extrapolation assessment, highlighting the potential of knowledge-graph-augmented LLMs as reliable tools for extrapolative modeling in additive manufacturing.