Radon Spectral Neural Operator: Learning PDE Operators in the Radon-Fourier Domain

Standard Fourier neural operators use frequency-wise multipliers in their linear spectral branches. We introduce the Radon Spectral Neural Operator (RSNO), a hybrid neural operator that combines multiscale spatial processing with structured global interactions in Radon-Fourier coordinates. The spatial branch aggregates fine- and coarse-resolution convolutional features, while the spectral branch applies orientation-dependent channel filtering and cross-orientation mixing within radial frequency shells. Their outputs are fused in each residual block, making multiscale aggregation and global spectral processing the two central architectural components. We prove an exact shellwise representation of the continuous branch, establish Sobolev stability, and show that different angular actions range from standard Fourier multipliers to nonlocal Fourier-integral structures. Across four PDE benchmarks spanning Darcy flow, Navier-Stokes forecasting, Helmholtz waves, and Burgers dynamics, RSNO delivers consistently strong performance against a broad set of neural-operator baselines. Ablations show that removing the Radon spectral branch increases mean error across all evaluated benchmark settings, whereas removing cross-orientation coupling produces substantially smaller changes. Together, these results support shellwise Radon–Fourier processing as a principled and analyzable global component within a multiscale neural-operator architecture.