算子学习的神经均值与核修正方法

中文摘要 AI 辅助

我们将神经网络均值与其残差的精确Matérn核回归、以及其学习到的特征相结合，并在两个公开的仿真问题上评估该组合方法，与已发表的基准方法对比：de Hoop等人的结构力学基准问题，以及Lamminpää等人的OCO-2辐射传输仿真器。在结构力学问题上，该组合方法达到4.55%的测试误差，与已发表的最优架构相当；在低数据 regime下，其测试误差为5.38%，优于已发表的6.49%。在OCO-2问题上，该组合方法在该问题自身的测试点上优于已发表的高斯过程仿真器，在三个光谱带中有两个直接胜出；在原始状态上落后网络十倍的同一核，在网络特征上反超了网络，我们测量了原因（在固定有效维度下，目标的原生空间平方范数下降约四十倍）并证明了该机制。当两类方法表现持平的场景中，我们训练的每个架构的残差相关性均超过0.86，且它们的共享成分在多样性和样本量上保持稳定，这表明已发表的性能 plateau是数据的属性。支持结果包括一个可从测量的相关性预测集成结果的二阶矩恒等式、一个最优恢复证书，以及一个无分布覆盖带——这是在我们的测试中唯一留存的不确定性信号。

英文摘要

We combine neural network means with exact Matérn kernel regressions of their residuals and of their learned features, and evaluate the pairing on two public emulation problems with published baselines: the structural-mechanics benchmark of de Hoop et al. and the OCO-2 radiative-transfer emulator of Lamminpää et al. On structural mechanics the combination reaches 4.55% test error, matching the best published architecture, and 5.38% against a published 6.49% in the low-data regime. On OCO-2 it improves on the published Gaussian-process emulator on that problem's own test points, outright on two of the three spectral bands; the same kernel that trails the network tenfold on the raw state overtakes it on the network's features, and we measure why (the target's squared native-space norm drops about fortyfold at fixed effective dimension) and prove the mechanism. Where the two families tie instead, the residuals of every architecture we train correlate above 0.86 and their shared component is flat in diversity and sample size, which reads the published plateau as a property of the data. Supporting results include a second-moment identity that predicts stacking outcomes from measured correlations, an optimal-recovery certificate, and a distribution-free coverage band, the only uncertainty signal that survives our tests.

发表机构

• Einstein Institute of Mathematics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）

机构由 AI 辅助整理，请以论文原文为准。