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

Abstract

Dirichlet-to-Neumann (DtN) maps send boundary values of a partial differential equation (PDE) solution to its normal derivative on the boundary. Learning such maps across varying domains is important for boundary-value problems, but a black-box neural operator must model both the operator's non-smoothing principal behavior and its dependence on boundary geometry. We use the boundary integral representation of the DtN map to obtain, for smooth planar boundaries, a useful principal-part decomposition: a geometry-independent leading operator can be written as a universal Fourier multiplier, while the remaining geometry-dependent correction is smoother. We propose Principal-Part Decomposed Neural Operators (PPDNO), a hybrid analytic-neural framework that turns this decomposition into a geometry-conditioned operator learning model. PPDNO computes the principal part by FFT and trains a low-rank Deep Operator Network (DeepONet)-type architecture to approximate only the residual correction across families of boundary geometries. This design keeps the exact linear action on the boundary data, exposes the sampled boundary as an input to the model, and turns the learned target into a smoother operator family. We justify the decomposition theoretically by proving smoothing and separated-approximation properties of the residual, and we derive finite-node and training-error bounds for the reconstructed full map. Experiments on interior Laplace problems over elliptical and Fourier-parameterized domains, and on exterior Helmholtz problems over rose curves, show that PPDNO improves accuracy over direct neural operator baselines while adding little inference overhead and generalizing to unseen boundary data. These results suggest that analytic operator structure and geometry-conditioned learning can be combined effectively for boundary solution maps.