Detecting and Discriminating Operator Misspecification in Hybrid PDE-Parameter Learning: a Reference-Free Instrument, with Discrimination Bounded In Sample

混合偏微分方程（PDE）参数学习中算子误设的检测与区分：一种无参考工具及其样本内界限

Abstract: We build an instrument that reads, from a single fit and with no oracle, whether the operator a hybrid PDE-parameter estimator postulates is wrong—and separates that from a merely unidentifiable parameter. 摘要： 我们构建了一种工具，无需预言机（oracle），仅通过单次拟合即可读取混合偏微分方程（PDE）参数估计器所假设的算子是否错误，并将其与仅仅是参数不可识别的情况区分开来。

On one self-adjoint parabolic inverse problem, an information-matrix statistic with plug-in scale and per-seed parameter has median 0.19 under correct specification, rejection rate $0.033$ against a pre-registered ceiling of $0.10$, and rises to $224$ and $85$ under two misspecifications, firing in every replicate. 在一个自伴随抛物线反问题中，使用插件尺度（plug-in scale）和逐种子参数的信息矩阵统计量在正确设定下中位数为 0.19，拒绝率为 $0.033$（预设上限为 $0.10$）；而在两种误设情况下，该统计量分别升至 $224$ 和 $85$，且在每次重复实验中均能触发。

On a correctly specified but non-identifiable design it stays mute—$0.050$ at $n=200$, Clopper-Pearson $[0.024, 0.090]$—while a rank statistic collapses to zero at a pre-registered boundary $c_5^=2.15\times10^{-3}.$ Two readings of one fit therefore separate the two failures across the three designs a deployable test reaches. 在设定正确但不可识别的设计中，该工具保持静默——在 $n=200$ 时为 $0.050$，Clopper-Pearson 区间为 $[0.024, 0.090]$——同时，秩统计量在预设边界 $c_5^=2.15\times10^{-3}$ 处坍缩为零。因此，通过单次拟合的两次读数，可以在可部署测试所涵盖的三种设计中区分这两种失效情况。

That separation is the contribution; detection alone is a crowded flank. In sample it is a bound, out of sample a direction. It is needed because the usual accuracy check is blind: the misspecified estimator’s in-domain RMSE is $2.7\times 10^{-2}$, below the observation noise for $\sigma\geq 0.05,$ while the coefficient is wrong by $29.7%$ at zero noise, $31.2%$ at the loudest. 这种区分能力正是本文的贡献所在；单纯的检测已是竞争激烈的领域。在样本内它是一个界限，在样本外它是一个方向。之所以需要它，是因为常规的精度检查是盲目的：误设估计器的域内 RMSE 为 $2.7\times 10^{-2}$，低于 $\sigma\geq 0.05$ 的观测噪声，而其系数在零噪声下偏差达 $29.7%$，在最大噪声下偏差达 $31.2%$。

Nor is the failure architectural: a one-parameter curve fit, a bare parameter and multilayer perceptrons of $49$ and $241$ parameters converge to the same pseudo-true, matched in closed form to $0.07%,$ whereas a physics-informed network, with its composite objective, converges to a disjoint one. 这种失效并非源于架构：单参数曲线拟合、裸参数以及参数量分别为 $49$ 和 $241$ 的多层感知机均收敛于相同的伪真值（闭式匹配误差为 $0.07%$），而具有复合目标的物理信息神经网络（PINN）则收敛于一个不相交的值。

We report where the instrument is blind, a pre-registered negative where a neural estimator loses to Tikhonov-regularized inversion at recovery, and the hypothesis under which its guarantee holds but a trained network violates it. 我们报告了该工具失效的场景，即一个预注册的负面案例：神经估计器在恢复任务中败给了 Tikhonov 正则化反演，并指出了在该假设下其保证成立，但训练后的网络却违反了该保证的情况。