We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which admit discrete spectral decompositions, and (ii) diagonal kernels of the form $K(x,x')=k(x,x')T$, where $k$ is a scalar-valued kernel and $T$ is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical $K=kI$ paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish dimension-free polynomial convergence rates, demonstrating that nonlinear operator learning can overcome the curse of dimensionality. The use of general operator-valued kernels further allows us to derive rates for intrinsically nonlinear operator learning, going beyond the linear-type behavior inherent in diagonal constructions of $K=kI$. Importantly, this framework accommodates a wide range of operator learning tasks, ranging from integral operators such as Fredholm operators to architectures based on encoder-decoder representations. Moreover, we validate its effectiveness through numerical experiments on the two-dimensional Navier-Stokes equations.

翻译：我们开发了一个随机逼近框架，用于利用一般的Mercer算子值核学习无限维空间之间的非线性算子。我们的框架涵盖两个关键类别：(i) 紧核，其允许离散谱分解；(ii) 对角核，形式为$K(x,x')=k(x,x')T$，其中$k$是标量值核，$T$是输出空间上的正算子。这一广泛设定诱导了富有表现力的向量值再生核希尔伯特空间（RKHSs），其推广了经典的$K=kI$范式，从而使得具有严格理论保证的丰富结构建模成为可能。为了处理位于RKHS之外的目标算子，我们引入了向量值插值空间来精确量化设定误差。在此框架内，我们建立了与维度无关的多项式收敛速率，证明了非线性算子学习能够克服维度灾难。使用一般的算子值核进一步使我们能够推导出本质非线性算子学习的速率，超越了$K=kI$对角构造中固有的线性类型行为。重要的是，该框架适用于广泛的算子学习任务，范围从积分算子（如Fredholm算子）到基于编码器-解码器表示的架构。此外，我们通过在二维Navier-Stokes方程上的数值实验验证了其有效性。