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This paper introduces mathematical invariant-enabled topological neural networks (MITNNs) to address limitations in existing molecular and materials learning by representing complex 3D structures through multiple complementary mathematical views integrated with topological neural architectures. By combining multiscale invariants from diverse mathematical fields, MITNNs capture richer structural information than single-representation methods. Systematic analyses reveal that specific pairings of invariant subsets and neural architectures yield superior predictive performance, consistently outperforming existing methods across various scientific prediction tasks.
Unlocking superior molecular and materials property prediction by fusing diverse mathematical invariants with topological neural networks, revealing that optimal performance hinges on carefully curated representation-architecture pairings.
Existing molecular and materials learning approaches often rely on a limited set of structural representations, which may capture only selected aspects of complex three-dimensional structure. Here, we introduce mathematical invariant-enabled topological neural networks (MITNNs), a framework that represents complex structures through multiple complementary mathematical views and integrates them with topological neural architectures. MITNNs combine multiscale invariants from topology, spectral theory, commutative algebra, differential geometry, and discrete curvature, capturing complementary structural information from the same system. Systematic invariant-subset, architecture-subset, and ensemble analyses show that predictive performance depends on how mathematical representations and neural architectures are paired, with selected combinations outperforming individual models and the aggregation of all available components. Across protein-ligand binding, metal-organic framework properties, mutation-induced protein solubility, and molecular toxicity prediction, MITNN consistently outperforms existing methods. These results establish MITNN as a mathematically multimodal framework for scientific machine learning.