Learning Green's Operators with Separable Modulated Kernels

Neural Green's operators learn a reusable kernel that specifies how a unit source at one point contributes to a linear PDE's solution at another. Existing basis expansions combine spatial functions that remain fixed as the source point moves, which can require many terms to resolve local features. Motivated by fundamental solutions, we introduce the Separable Modulation Neural Operator (SMNO). We represent the Green's function as a sum of learned offset kernels, each multiplied by separate spatial weights at the source and evaluation points. The kernels depend only on the offset between these points and are shared across PDE instances. Both sets of weights are predicted from the domain shape and the coefficient field. On uniform grids, we apply the operator with fast Fourier transforms (FFTs) without constructing a dense kernel matrix. On Poisson problems with unseen 2D and 3D geometries, SMNO reduces the error on higher-frequency sources by 71–94% relative to basis-expansion Green's operators, and on Darcy flow with unseen coefficient fields it reduces the same error by 82% relative to FNO. A variant with a learned boundary map reduces NGF's error on the Mechanical Components Benchmark by 41%. On a 341×341 grid, building the operator and applying it to 64 sources takes 69 ms, versus 135 ms for FNO, 2.1 s for Transolver, and 10 s for a dense pairwise kernel.