We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator $\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V}$ are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations $\phi(u_i), \varphi(v_i)$ of input/output functions $v_i=\mathcal{G}^\dagger(u_i)$ ($i=1,\ldots,N$), and the measurement operators $\phi\,:\, \mathcal{U}\to \mathbb{R}^n$ and $\varphi\,:\, \mathcal{V} \to \mathbb{R}^m$ are linear. Writing $\psi\,:\, \mathbb{R}^n \to \mathcal{U}$ and $\chi\,:\, \mathbb{R}^m \to \mathcal{V}$ for the optimal recovery maps associated with $\phi$ and $\varphi$, we approximate $\mathcal{G}^\dagger$ with $\bar{\mathcal{G}}=\chi \circ \bar{f} \circ \phi$ where $\bar{f}$ is an optimal recovery approximation of $f^\dagger:=\varphi \circ \mathcal{G}^\dagger \circ \psi\,:\,\mathbb{R}^n \to \mathbb{R}^m$. We show that, even when using vanilla kernels (e.g., linear or Mat\'{e}rn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.

翻译：我们提出了一种基于核的通用框架，用于学习巴拿赫空间之间的算子，并附带了先验误差分析以及与深度算子网络和傅里叶神经算子等流行神经网络方法的全面数值比较。我们考虑目标算子的输入/输出空间为再生核希尔伯特空间（RKHS）的情况，数据以输入/输出函数的部分观测形式给出，且测量算子为线性。定义与相关的优化恢复映射，我们通过来近似，其中是的优化恢复近似。结果表明，即使使用普通核（如线性核或Matérn核），我们的方法在成本-精度权衡方面具有竞争力，且在大多数基准测试中与神经网络方法性能相当或更优。此外，我们的框架继承核方法的若干优势：简洁性、可解释性、收敛保证、先验误差估计以及贝叶斯不确定性量化。因此，它可作为算子学习的自然基准。