FFT-Free Neural Operators for Helmholtz Scattering via Adaptive Coefficient Modulation

Fourier Neural Operators (FNOs) exhibit mode saturation on high-contrast inhomogeneous media, and recent multi-scale extensions (MscaleFNO) further worsen out-of-distribution (OOD) generalization. We introduce the Helmholtz Neural Operator (HNO), a physics-informed, FFT-free branch–trunk operator in the DeepONet family, with a hybrid SIREN+learnable-Fourier trunk and a dual-path rank-32 hypernetwork branch, with bounded multiplicative gating on per-mode coefficients. At a matched parameter count (∼1.05 M, five seeds), HNO achieves a 2.6× lower OOD generalization gap than FNO (19.6% vs. 50.6%, $p=1.7\times {10}^{-3}$, Cohen’s $d=5.1$), 5.1× lower than vanilla DeepONet (19.6% vs. 99.9%, $p=8.2\times {10}^{-3}$), and 6.0× lower than MscaleFNO (19.6% vs. 117.4%, $p=2.4\times {10}^{-6}$); MscaleFNO’s deficit grows at 4.2× more parameters, ruling out capacity starvation. HNO is 4.6×/16.4× faster than FNO/MscaleFNO and 64×–245× faster than multi-threaded FD-PML (MKL PARDISO, 12 cores; 183×–698× vs. single-thread scipy.spsolve), making it suitable as a forward surrogate inside many-query workflows. Absolute accuracy on extreme-contrast (15:1) OOD samples is limited (relative $L^2\approx 1$), so HNO is positioned as a many-query surrogate or warm start for refinement loops, not a stand-alone replacement for direct solvers. A scope limitation is that HNO underperforms FNO on elliptic Darcy Flow, confirming specialization for hyperbolic/wave equations rather than universal operator learning.