Title: Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

Abstract:

This study presents the Kernel Neural Operator (KNO), an architecture designed for operator learning that is provably convergent. It employs compositions of deep kernel-based integral operators to approximate functions in function space, effectively mapping functions to other functions. A key feature of the KNO is its decoupling of kernel selection from the numerical integration scheme, or quadrature. This separation enables operator learning on irregular geometries using explicitly chosen trainable kernels. For such irregular domains, the KNO can apply quadrature rules tailored to the specific geometry. To address the challenges posed by the curse of dimensionality, the method incorporates an efficient dimension-wise factorization algorithm when dealing with regular domains. Furthermore, the capacity to define kernels explicitly permits the use of highly expressive, non-stationary, neural anisotropic kernels, with parameters determined through neural network training. We establish universal approximation theorems, demonstrating that both the continuous and fully discretized forms of the KNO serve as universal approximators for operator learning tasks. Empirical evaluations on standard benchmarks reveal that KNOs achieve training and test accuracies that are comparable to or superior to those of established neural operators, despite typically requiring an order of magnitude fewer trainable parameters. The enhanced expressiveness of the kernels is shown to be critical for achieving high accuracy. Consequently, KNOs enable deep operator learning that is low-memory and geometrically flexible, while maintaining the implementation simplicity and transparency characteristic of traditional kernel methods in both scientific computing and machine learning.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC