论文把算子学习的误差分解讲得很清楚，还给出了N、n、m的预算分配条件，物理信息扩展省去重新训练，做相关研究值得细读。
该论文提出两阶段采样框架，离线核回归算子从N个输入-输出对学习算子的离散表示，在线核重构算子从预测观测恢复输出函数。理论贡献是推导了预算分配条件，要求训练对数量N、输入观测数n和输出分辨率m满足耦合关系以保证收敛。总误差分解为重构误差和学习误差，可独立分析，并得到定量缩放律。此外，引入物理信息扩展，在线重构时通过惩罚PDE残差增强约束，无需重新训练。数值实验验证了理论扩展的有效性。
Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extension
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.