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).
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2603.16849 [cs.LG] |
| (or arXiv:2603.16849v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2603.16849 arXiv-issued DOI via DataCite |
From: Mattia Rigotti [view email]
[v1]
Tue, 17 Mar 2026 17:54:26 UTC (168 KB)
[v2]
Fri, 10 Apr 2026 22:03:26 UTC (171 KB)
[v3]
Tue, 12 May 2026 15:12:26 UTC (136 KB)
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