This paper introduces the Kernel Neural Operator (KNO), a novel operator learning technique that uses deep kernel-based integral operators in conjunction with quadrature for function-space approximation of operators (maps from functions to functions). KNOs use parameterized, closed-form, finitely-smooth, and compactly-supported kernels with trainable sparsity parameters within the integral operators to significantly reduce the number of parameters that must be learned relative to existing neural operators. Moreover, the use of quadrature for numerical integration endows the KNO with geometric flexibility that enables operator learning on irregular geometries. Numerical results demonstrate that on existing benchmarks the training and test accuracy of KNOs is higher than popular operator learning techniques while using at least an order of magnitude fewer trainable parameters. KNOs thus represent a new paradigm of low-memory, geometrically-flexible, deep operator learning, while retaining the implementation simplicity and transparency of traditional kernel methods from both scientific computing and machine learning.

翻译：本文提出核神经算子(KNO)，一种新颖的算子学习技术，通过结合深度核积分算子与数值积分方法，实现函数到函数映射的泛函空间逼近。KNO在积分算子中采用参数化、闭式、有限光滑且紧支撑的核函数，并配备可训练的稀疏参数，相较于现有神经算子方法显著减少了需学习的参数量。此外，数值积分方法的使用赋予KNO几何灵活性，使其能在不规则几何域上进行算子学习。数值实验表明，在现有基准测试中，KNO的训练与测试精度均优于主流算子学习方法，同时使用的可训练参数量至少降低一个数量级。因此，KNO代表了一种低内存、几何灵活、深度算子学习的新范式，同时保持了科学计算与机器学习领域传统核方法在实现简洁性与理论透明性方面的优势。