Graph Neural Networks for the Analysis of Complex Mathematical Structures

Graph Neural Networks (GNNs) have emerged as a powerful framework for analyzing complex mathematical structures that can be represented as graphs, including algebraic objects, geometric spaces, and combinatorial structures. This comprehensive review examines the theoretical foundations, architectural variants, and applications of GNNs to mathematical analysis, highlighting how these networks effectively capture the relational inductive biases inherent in mathematical systems. We explore how GNNs generalize traditional neural network operations to graph-structured data through message passing, aggregation, and update mechanisms, enabling them to process mathematical structures with irregular and non-Euclidean properties. The paper surveys specialized GNN architectures including spectral convolutional networks, attention-based models, and geometrically equivariant networks that preserve mathematical symmetries. Through detailed case studies in group theory, topology, combinatorics, and mathematical physics, we demonstrate how GNNs can discover patterns, predict properties, and generate conjectures about mathematical structures. The review also addresses current challenges and limitations while outlining promising research directions for advancing the integration of graph representation learning with mathematical reasoning.