Is turbulence really like Jello-O? Science backs up the goofy analogy. The viral TikTok video may actually hold up under scrutiny. Breakthroughs, discoveries, and DIY tips sent six days a week. A young woman pushes a balled-up piece of napkin into a cup of Jell-O, asking the viewer to imagine that it is an airplane, high in the air.

How pilots avoid thunderstorms--and what happens when they can't Most commercial planes get struck by lightning a couple times a year. Despite the fears of nervous fliers, radar, routing, and teamwork keep planes safe during storms. Breakthroughs, discoveries, and DIY tips sent every weekday. In the 2023 movie starring Gerard Butler, a commercial aircraft is caught in a terrible storm. The plane shakes and the lights go out.

The dominant paradigm for learning on graph-structured data is message passing. Despite being a strong inductive bias, the local message passing mechanism suffers from pathological issues such as over-smoothing, over-squashing, and limited node-level expressivity. To address these limitations we propose Bundle Neural Networks (BuNN), a new type of GNN that operates via message diffusion over flat vector bundles - structures analogous to connections on Riemannian manifolds that augment the graph by assigning to each node a vector space and an orthogonal map. A BuNN layer evolves the features according to a diffusion-type partial differential equation. When discretized, BuNNs are a special case of Sheaf Neural Networks (SNNs), a recently proposed MPNN capable of mitigating over-smoothing. The continuous nature of message diffusion enables BuNNs to operate on larger scales of the graph and, therefore, to mitigate over-squashing. Finally, we prove that BuNN can approximate any feature transformation over nodes on any (potentially infinite) family of graphs given injective positional encodings, resulting in universal node-level expressivity. We support our theory via synthetic experiments and showcase the strong empirical performance of BuNNs over a range of real-world tasks, achieving state-of-the-art results on several standard benchmarks in transductive and inductive settings.