Differentiable Čech Nerves: Learning Covers that Optimize Cohomology with Grothendieck GNNs

This project advances Grothendieck Graph Neural Networks (GGNNs) by making the nerve-of-cover construction differentiable. It parameterizes covers and sieves, builds their nerves, computes Čech cohomology (and sheaf cohomology for labeled data), and incorporates losses that align cohomology with ground-truth topology or task-derived constraints. This creates an end-to-end pipeline optimizing graph representations to preserve or enforce topological invariants. The approach connects directly to safety-oriented coverage metrics using algebraic topology, enabling models to satisfy coverage criteria defined via homology of decision regions or specification sheaves. Theoretically, the layer is framed via homotopy colimits and higher structures, providing stability conditions for cohomology under small cover changes. The novelty lies in learning the cover itself via Grothendieck-style sieves rather than post hoc invariant computation, closing the loop between representation, topology, and task objectives.

References:

1. Grothendieck Graph Neural Networks Framework: An Algebraic Platform for Crafting Topology-Aware GNNs. A. S. Langari, Leila Yeganeh, Kim Khoa Nguyen (2024). arXiv.org.
2. Coupling algebraic topology theory, formal methods and safety requirements toward a new coverage metric for artificial intelligence models. Faouzi Adjed, Mallek Mziou-Sallami, Frédéric Pelliccia, Mehdi Rezzoug, L. Schott, Christopher A. Bohn, Yesmina Jaâfra (2022). Neural computing & applications (Print).