用拓扑与上同调流构建类脑表征学习框架，提升模型稳定性与抗噪能力。
Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning
- 将神经计算建模为动态单纯复形上的上链映射演化
- 在合成与真实神经数据上均实现更高流形一致性和抗噪性
- 适合对拓扑驱动学习、脑启发模型感兴趣的科研人员
本文提出一个基于持久同调结构与上同调流相互作用的数学严谨类脑表征学习框架。神经计算被重新定义为动态单纯复形上的上链映射演化，从而捕捉时空与功能脑状态中的不变量。该架构结合代数拓扑与微分几何，构建广义梯度学习的上同调算子，运行于同调景观中。通过持久同调、层上同调与谱拉普拉斯算子，联合分析具有可控拓扑特征的合成数据及真实神经数据，量化其稳定性、连续性与结构保真度。实验表明，该模型在流形一致性与抗噪能力方面优于图神经网络和基于流形的深度架构，为拓扑驱动的表征学习提供了统一的数学基础。
原文摘要 · Abstract (English)
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.
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