Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension

Jul 16, 2026

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Kernel-based operator learning is analyzed in a two-stage sampling framework combining offline kernel regression that learns a discretized target operator from input-output pairs with an online kernel reconstruction that recovers the output function from predicted observations. A central result is an explicit budget allocation condition linking the number N of training pairs, the number n of input observations, and the output resolution m. Coupled error analysis decomposes total error into reconstruction and learning terms, yielding scaling laws that guarantee convergence. A physics-informed extension penalizes PDE residuals during online reconstruction at collocation points, enabling improved accuracy without retraining.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

Introduction

Two-stage kernel operator learning framework

Theoretical contribution: budget allocation and error decomposition

Physics-informed extension with PDE residual penalization

arXiv :2607 .06287v1 [math .NA] 7 Jul 2026 Kernel-based Operator Learning Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension R¨udiger Kempf [ruediger.kempf@uni-bayreuth.de](ruediger.kempf@uni-bayreuth.de) Applied and Numerical Analysis University of Bayreuth 95440 Bayreuth, Germany Editor: Abstract We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number N of training pairs, the number n of input observations, and the output resolution m. The condition is derived from a coupled error analysis that interprets the surrogate as a reconstruction from approximate data. This yields a decomposition of the total error into reconstruction and learning contributions that can be analyzed independently. As a consequence, we obtain quantitative scaling laws describing how N , n, and m must be coupled to guarantee convergence and to balance offline learning and online reconstruction errors. The resulting estimates extend previous analyses of kernel-based operator learning. We further introduce a physics-informed extension that incorporates knowledge of the underlying PDE at evaluation time. Rather than encoding constraints directly into the kernel, we augment the online reconstruction step by penalizing PDE residuals at collocation points. The method requires no retraining for new inputs. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the proposed physics-informed reconstruction strategy. Keywords: kernel methods, operator learning, budget allocation, physics-informed learning 1 Introduction Operator learning has emerged as a central problem in scientific machine learning: given a finite collection of input-output pairs generated by an unknown operator G : U → V between function spaces, the goal is to construct a surrogate A ≈ G that generalizes to unseen inputs. The canonical motivation is the solution operator of a PDE, where G maps a coefficient or forcing function to the corresponding solution, and one wishes to replace expensive repeated PDE solves with a fast surrogate at deployment time. The dominant paradigm for operator learning is neural-network-based. Architectures such as DeepONet (Lu et al., 2021, 2022; Wang et al., 2021, 2022) and the Fourier Neural Operator (Li et al., 2021) have achieved impressive empirical performance on a range of benchmarks. However, neural-network approaches offer limited theoretical guarantees: R. Kempf convergence rates, sample complexity, and the influence of design parameters such as training set size and output resolution are difficult to characterize rigorously. Kernel methods, by contrast, come with a well-developed approximation theory (Wendland, 2004; Schaback and Wendland, 2006; Owhadi and Scovel, 2019; Sch¨olkopf and Smola, 2001), deterministic error bounds (Narcowich et al., 2005; Arcangeli et al., 2012; Duchon, 1978), and transparent dependence on all design parameters, such as the correlation length of the kernel (Wendland and Rieger, 2005; Sun and Wang, 2026) . While classical kernel methods are well understood in the finite-dimensional function approximation setting, their extension to operator learning is more recent and mainly of numerical nature (Kadri et al., 2016; Owhadi, 2023; Batlle et al., 2024; Sharma et al., 2026; Nelsen and Stuart, 2024) . This paper contributes to the rigorous, approximation-theoretic foundation of kernel-based operator learning. The framework we study builds on the two-stage approach of Batlle et al. (2024); Sharma et al. (2026), in which an offline kernel regression operator A off learns

FAQs

How does the two-stage framework work in kernel-based operator learning?

Offline kernel regression learns a discretized representation of the target operator from training input-output pairs. Online kernel reconstruction then recovers the output function from predicted observations.

What is the key theoretical result about budget allocation?

An explicit budget allocation condition links N training pairs, n input observations, and output resolution m. It is derived via coupled error analysis that decomposes total error into reconstruction and learning contributions.

How is physics information incorporated at evaluation time?

The method adds a physics-informed penalty by computing and minimizing PDE residuals at collocation points during the online reconstruction step. It avoids retraining for new inputs.

Jul 16, 2026

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