{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/kernel-methods-are-competitive-for-operator","title":"Kernel Methods are Competitive for Operator Learning","arxiv_id":"2304.13202","date":"2023-04-26","proceeding":null,"authors":["Pau Batlle","Matthieu Darcy","Bamdad Hosseini","Houman Owhadi"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2304.13202v2","url_pdf":"https://arxiv.org/pdf/2304.13202v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"kernel-methods-are-competitive-for-operator","repo_url":"https://github.com/matthieudarcy/kernelsoperatorlearning","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"operator-learning","task_name":"Operator learning"},{"task_slug":"uncertainty-quantification","task_name":"Uncertainty Quantification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2304.13202","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}