A Graph-Based Framework for Exploring Mathematical Patterns in Physics: A Proof of Concept

The vast corpus of physics equations forms an implicit network of mathematical relationships that traditional analysis cannot fully explore. This work introduces a graph-based framework combining neural networks with symbolic analysis to systematically discover and validate mathematical patterns across physics domains. Starting from 659 equations, we performed rigorous semantic disambiguation to resolve notational polysemy affecting 213 equations, then focused on 400 advanced physics equations by excluding elementary mechanics to emphasize inter-branch connections of modern physics. This corpus was represented as a weighted knowledge graph where a Graph Attention Network achieved 97.4% AUC in link prediction, significantly outperforming classical baselines. The framework's primary value emerges from its dual capability: generating hypotheses and auditing knowledge. First, it functions as a hypothesis generator, producing hundreds of candidate cross-domain connections, from blackbody radiation coupled with Navier-Stokes equations to radioactive decay linked with electromagnetic induction. Second, through symbolic analysis of 30 equation clusters, it serves as a computational auditor that verified established theory consistencies, synthesized the Magnetic Reynolds Number from electromagnetic-fluid coupling, and revealed how even parsing errors could potentially point toward legitimate research like analog gravity. This proof-of-concept intentionally over-generates candidates to ensure comprehensive exploration of mathematical possibility space. Even tautologies and errors serve scientific purposes: redundancy identification and knowledge base quality assessment. The system transforms the intractable combinatorial space into a filtered stream of mathematical patterns for human interpretation.