MF-SPECBO: MULTI-FIDELITY SPECTRAL BAYESIAN OPTIMIZATION FOR NEURAL OPERATORS

Under Direct Spectral Evaluation (DSE), a neural operator’s cost is set by which Fourier modes it keeps, not just how many — yet whether informed mode selection can beat a plain isotropic truncation is untested. We give the question a screen: the gain of any selection rule over the matched-budget radial box is at most the spectrum’s radial defect ∆, a one-pass statistic (Theorem 1). We then build the machinery to search — a Sobolev chordal set kernel, a downward-closed restriction small enough for exact acquisition maximization, a calibrated MF-GP-UCB fidelity rule — and run the comparison at exactly matched cardinality. On standard benchmarks ∆ is tiny and, as predicted, no choice of mode set beats the box (a growth curriculum on the same set does — the win is in training, not selection); compression still yields 86.2–90.7% fewer modes, with 2.59–2.68× measured speedup on meshes above 103 nodes (the smallest mesh is launch-bound and gains none), but needs no search. On an anisotropic Darcy-flux family that passes the screen, MF-SpecBO improves on the box by up to 11.55 points and on a one-pass energy heuristic by up to 9.70 (p < 0.001 at κ=4), losing the advantage again at extreme anisotropy where the winning modes exceed what 800 samples can estimate. Mode selection pays in a regime ∆ locates in advance; every tabulated number is machine-verified against its recorded measurement.