Principal-Part Decomposition for Neural Operator Learning of Dirichlet-to-Neumann Maps

Dirichlet-to-Neumann (DtN) maps encode PDE responses on the boundary by mapping Dirichlet data to Neumann data. Learning these operators across varying geometries is challenging because DtN maps are nonlocal first-order operators with singular high-frequency behavior. For smooth planar boundaries, a boundary integral formulation decomposes the DtN map into a universal Fourier multiplier and a smoother geometry-dependent remainder. We introduce Principal-Part Decomposed Neural Operators (PPDNO), a hybrid analytic-neural framework that computes the principal part exactly via FFT and learns only the remainder using a geometry-conditioned low-rank network that preserves linearity in the boundary data. We prove smoothing and separated-approximation properties of the remainder and derive error bounds for the reconstructed map. Experiments on Laplace and Helmholtz problems show that PPDNO improves accuracy and generalization over standard neural operators and residualized baselines while maintaining high inference efficiency.