[2603.14831] Neural Networks as Local-to-Global Computations

Mathematics > Algebraic Topology arXiv:2603.14831 (math) [Submitted on 16 Mar 2026 (v1), last revised 24 Mar 2026 (this version, v3)] Title:Neural Networks as Local-to-Global Computations Authors:Vicente Bosca, Robert Ghrist View a PDF of the paper titled Neural Networks as Local-to-Global Computations, by Vicente Bosca and Robert Ghrist View PDF Abstract: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 t...