物理信息核方法的快速学习率

中文摘要 AI 辅助

在物理信息机器学习中，目标函数 $u^*$ 从带噪声的值观测 $y_i=u^*(x_i)+ \varepsilon_i$ 以及微分信息中学习，微分信息由带噪声的观测 $d_j=(Du^*)(z_j)+\xi_j$ 或已知的物理约束 $Du^*=v$ 给出。我们考虑 $D$ 为线性微分算子的情形，并分析一种结合 $n$ 个值观测和 $m$ 个微分观测的物理信息核估计器 $\hat u$。在此背景下，我们询问微分信息能在多大程度上改善预测，以及这种改善如何定量地依赖于 $n$、$m$ 和 $D$。我们证明了有限样本界，并由数值模拟支持，揭示了预测误差的两阶段结构。当 $m$ 有限时，学习率同时依赖于 $n$ 和 $m$；当 $m$ 超过一个依赖于问题的阈值时，学习率饱和并达到在施加完美约束 $D \hat u = Du^*$ 时获得的oracle率。我们讨论了Sobolev空间的例子，这些空间是再生核希尔伯特空间，包括环面上的部分拉普拉斯约束和有界域上的梯度观测。这些例子展示了可能的学习率改善范围——从标准的非参数率 $n^{-1/4}$ 到参数率 $n^{-1/2}$。最后，我们在一个更强的范数中推导出物理一致的学习率，该范数联合控制 $\hat u$ 和 $D\hat u$ 的误差。

英文摘要

In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential operator and analyze a physics-informed kernel estimator $\hat u$ combining $n$ value observations and $m$ differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on $n$, $m$, and $D$. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When $m$ is limited, the rate depends jointly on $n$ and $m$; when $m$ exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint $D \hat u = Du^*$ is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric $n^{-1/4}$ to the parametric rate $n^{-1/2}$. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in $\hat u$ and $D\hat u$.

发表机构

• Istituto Italiano di Tecnologia（意大利理工学院）
• Università degli Studi di Genova（热那亚大学）

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