🤖 AI Summary
This work addresses the challenge of resource allocation among the number of training samples (N), input observation points (n), and output resolution (m) in operator learning. The authors propose a two-stage sampling framework: in the offline stage, a discrete representation of the operator is learned via kernel regression; in the online stage, the output function is reconstructed from predicted observations, enhanced by physics-informed constraints to improve accuracy. The study establishes a novel quantitative scaling law and error decomposition mechanism linking N, n, and m, and introduces a physics-informed online reconstruction strategy that avoids retraining. Theoretical analysis provides convergence guarantees and an error-balancing criterion, while numerical experiments validate the proposed scaling law and demonstrate the method’s superior performance in preserving physical consistency and achieving high reconstruction accuracy.
📝 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.