用层间拓扑结构解释神经网络计算，揭示前向传播的唯一性与可诊断性。
Neural Networks as Local-to-Global Computations
- 将神经网络转化为胞腔层化结构，每步计算对应边上的限制映射。
- 前向输出是边界数据的唯一调和延拓，且热方程指数收敛。
- 支持双向信息传播，可局部诊断、无反向传播训练，适合模型分析。
我们为任意前馈ReLU神经网络构建了一个胞腔层化结构：在前向传播中每个中间量设为一个顶点，每个计算步骤（仿射变换、激活、输出）作为边上的一条限制映射。受限上边界算子在自由坐标下为单位三角矩阵，其行列式恒为1，对任意激活模式，受限拉普拉斯算子均为正定。因此相对上同调为零，前向传播输出是边界数据的唯一调和延拓。尽管分段线性激活引入状态依赖切换，层间热方程仍指数收敛至该输出。与前向传播不同，热方程在层间双向传播信息，实现双向约束的固定神经元，通过局部偏差最小化完成训练而无需反向传播，并可按层与操作类型分解网络行为。我们在小型合成任务上验证了该框架，确认了收敛性定理，并证明基于层化结构的训练虽尚未超越随机梯度下降，但符合理论预测的定量缩放规律。
原文摘要 · Abstract (English)
We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a restriction map on an edge. The restricted coboundary operator on the free coordinates is unitriangular, so its determinant is $1$ and the restricted Laplacian is positive definite for every activation pattern. It follows that the relative cohomology vanishes and the forward pass output is the unique harmonic extension of the boundary data. The sheaf heat equation converges exponentially to this output despite the state-dependent switching introduced by piecewise linear activations. Unlike the forward pass, the heat equation propagates information bidirectionally across layers, enabling pinned neurons that impose constraints in both directions, training through local discrepancy minimization without a backward pass, and per-edge diagnostics that decompose network behavior by layer and operation type. We validate the framework experimentally on small synthetic tasks, confirming the convergence theorems and demonstrating that sheaf-based training, while not yet competitive with stochastic gradient descent, obeys quantitative scaling laws predicted by the theory.
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