从流形与不动点视角重解神经网络，揭示学习本质与架构设计新思路。
Deep Manifold Part 2: Neural Network Mathematics
- 用分段流形和不动点理论构建神经网络全局数学模型
- 指出能力涌现依赖于不动点区域的稳定，而非初始固定结构
- 为分布式弹性模型与世界建模提供几何与代数基础
本文通过堆叠的分段流形、不动点理论和边界条件迭代，推导出神经网络的全局方程。移除固定坐标与算子后，神经网络呈现为由流形复杂度、高阶非线性及边界条件塑造的可学习数值计算。真实数据带来强数据复杂性、近乎无限的范围、规模与小批量碎片化；训练动态则通过节点覆盖变化、曲率累积以及可塑性的生灭产生学习复杂性。这些因素共同制约可学习性，解释为何能力仅在不动点区域稳定时出现。神经网络并非始于不动点，而是通过残差驱动的迭代逐步构造。这一视角阐明了单体模型在几何与数据诱导的可塑性下的局限性，并推动将流形复杂性分布于多个弹性模型中，形成基于几何、代数、不动点与真实数据复杂性的连贯世界建模框架。
原文摘要 · Abstract (English)
This work develops the global equations of neural networks through stacked piecewise manifolds, fixed-point theory, and boundary-conditioned iteration. Once fixed coordinates and operators are removed, a neural network appears as a learnable numerical computation shaped by manifold complexity, high-order nonlinearity, and boundary conditions. Real-world data impose strong data complexity, near-infinite scope, scale, and minibatch fragmentation, while training dynamics produce learning complexity through shifting node covers, curvature accumulation, and the rise and decay of plasticity. These forces constrain learnability and explain why capability emerges only when fixed-point regions stabilize. Neural networks do not begin with fixed points; they construct them through residual-driven iteration. This perspective clarifies the limits of monolithic models under geometric and data-induced plasticity and motivates architectures and federated systems that distribute manifold complexity across many elastic models, forming a coherent world-modeling framework grounded in geometry, algebra, fixed points, and real-data complexity.
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