Geometric Generalization of Neural Operators from a Kernel Integral Perspective

Neural operators are neural network-based surrogate models for approximating partial differential equation solution operators, enabling efficient many-query computations in science and engineering when low-to-moderate accuracy is sufficient. Many applications, including engineering design, involve variable and often nonparametric geometries, for which generalization to unseen shapes remains a central practical challenge. In this work, we adopt a kernel-integral perspective motivated by classical boundary integral formulations and study operator learning on variable geometries through the approximation of geometry-dependent kernel operators, including singular kernels. This perspective clarifies a mechanism for geometric generalization for fixed linear operators and reveals a direct connection between operator learning and fast kernel summation methods. Leveraging this connection, we propose a multiscale point cloud neural operator inspired by Ewald summation, combining Fourier long-range interactions with local geometry-aware corrections. We further establish approximation guarantees for the resulting multiscale representation of linear operators defined by singular kernels. Numerical experiments demonstrate robust generalization across diverse geometries for several commonly used kernels, Laplace-type boundary integral maps, and a large-scale three-dimensional nonlinear vehicle flow example.