GIST: Gauge-Invariant Spectral Transformers for Scalable Graph Neural Operators

TL;DR

GIST introduces a scalable, gauge-invariant spectral transformer for graph neural operators, ensuring discretization invariance and achieving state-of-the-art results on large mesh benchmarks.

Contribution

It proposes a novel spectral attention mechanism that guarantees gauge invariance and provides a formal bound on discretization mismatch, a first in scalable graph neural operators.

Findings

GIST achieves state-of-the-art performance on large mesh benchmarks.

It maintains gauge invariance and discretization invariance across different graph resolutions.

GIST scales to graphs with up to 750K nodes while providing theoretical guarantees.

Abstract

Neural operators on irregular meshes face a fundamental tension. Spectral positional encodings, the natural choice for capturing geometry, require cubic-complexity eigendecomposition and inadvertently break gauge invariance through numerical solver artifacts; existing efficient approximations sacrifice gauge symmetry by design. Both failure modes break discretization invariance: models fail to transfer across mesh resolutions of the same domain, and similarly across different graphs of related structure in inductive settings. We propose GIST (Gauge-Invariant Spectral Transformer), a scalable neural operator that resolves this tension by restricting attention to pairwise inner products of efficient approximate spectral embeddings. We prove these inner products estimate an exactly gauge-invariant graph kernel at end-to-end $\mathcal{O}(N)$ complexity, and establish a formal connection between gauge invariance and discretization-invariant learning with bounded mismatch error. To our knowledge, GIST is the first scalable graph neural operator with a provable discretization-mismatch bound. Empirically, GIST sets state-of-the-art on the AirfRANS, ShapeNet-Car, DrivAerNet, and DrivAerNet++ mesh benchmarks (up to 750K nodes), and additionally matches strong baselines on standard graph benchmarks (e.g., 99.50% micro-F1 on PPI).