Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning
This provides a foundational mathematical approach for topology-driven representation learning in neuroscience and AI, though it appears incremental in applying established topological methods to this domain.
The paper tackles the problem of brain-inspired representation learning by developing a mathematical framework based on persistent topological structures and cohomological flows, achieving superior manifold consistency and noise resilience compared to existing architectures like graph neural networks.
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the evolution of cochain maps over dynamic simplicial complexes, enabling representations that capture invariants across temporal, spatial, and functional brain states. The proposed architecture integrates algebraic topology with differential geometry to construct cohomological operators that generalize gradient-based learning within a homological landscape. Synthetic data with controlled topological signatures and real neural datasets are jointly analyzed using persistent homology, sheaf cohomology, and spectral Laplacians to quantify stability, continuity, and structural preservation. Empirical results demonstrate that the model achieves superior manifold consistency and noise resilience compared to graph neural and manifold-based deep architectures, establishing a coherent mathematical foundation for topology-driven representation learning.