Geometric Generalization of Neural Operators from Kernel Integral Perspective

TL;DR

This paper introduces a kernel integral perspective for neural operators, enabling improved geometric generalization in solving parametric PDEs with variable geometries, supported by a multiscale neural operator and theoretical guarantees.

Contribution

It recasts operator learning as kernel approximation, connecting it with fast kernel methods, and proposes a multiscale neural operator with accuracy guarantees for diverse geometries.

Findings

Robust geometric generalization demonstrated across multiple kernels.

The proposed method achieves high accuracy in large-scale 3D fluid dynamics.

Theoretical guarantees support the approximation quality of the neural operator.

Abstract

Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications, including engineering design, involve variable and often nonparametric geometries, for which generalization to unseen geometries remains a central practical challenge. In this work, we adopt a kernel integral perspective motivated by classical boundary integral formulations and recast operator learning on variable geometries as the approximation of geometry-dependent kernel operators, potentially with singularities. This perspective clarifies a mechanism for geometric generalization and reveals a direct connection between operator learning and fast kernel summation methods. Leveraging this connection, we propose a multiscale neural operator inspired by Ewald summation for learning and efficiently evaluating unknown kernel integrals, and we provide theoretical accuracy guarantees for the resulting approximation. Numerical experiments demonstrate robust generalization across diverse geometries for several commonly used kernels and for a large-scale three-dimensional fluid dynamics example.