A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning

arXiv:2509.11070v4 Announce Type: replace-cross Abstract: 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) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable 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 non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.

Source: arXiv cs.LG | 2026-09-29