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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.11070 |
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| _version_ | 1866915719521763328 |
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| author | Yang, Jia-Qi Shi, Lei |
| author_facet | Yang, Jia-Qi Shi, Lei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning Yang, Jia-Qi Shi, Lei Machine Learning Numerical Analysis Functional Analysis Statistics Theory 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. |
| title | A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning |
| topic | Machine Learning Numerical Analysis Functional Analysis Statistics Theory |
| url | https://arxiv.org/abs/2509.11070 |