无需真值即可检测微分方程参数模型的算子错误，区分错误与不可识别问题。
Detecting and Discriminating Operator Misspecification in Hybrid PDE-Parameter Learning: a Reference-Free Instrument, with Discrimination Bounded In Sample
- 基于信息矩阵统计量，单次拟合即可判断算子是否误设。
- 算子正确时检验统计量中位数为0.19，错误时升至224；不可识别时保持静默。
- 适合关注模型可靠性、反问题与神经算子验证的研究者使用。
我们构建了一个无需真值参考的检测工具，仅通过一次拟合即可判断混合偏微分方程-参数估计器所假设的算子是否错误，并将此错误与参数不可识别区分开来。在一个自伴抛物型逆问题上，当算子正确时，信息矩阵统计量中位数为0.19，对预注册的0.10上限的拒绝率为0.033；在两种算子误设下，该统计量分别升至224和85，且在所有重复实验中均触发警报。在正确但不可识别的设计中，其值保持为0.050（n=200），Clopper-Pearson置信区间[0.024, 0.090]，而秩统计量则坍缩至预注册边界c₅*=2.15×10⁻³。单一拟合的两次读数可实现三种设计下的故障分离。该分离能力是核心贡献；仅检测错误属于已有研究密集领域。在样本内为有界约束，在样本外提供方向性信号。该工具必要，因常规精度检查盲于误设：即使系数误差达29.7%（零噪声）至31.2%（最大噪声），其域内RMSE仍低于σ≥0.05时的观测噪声水平2.7×10⁻²。非架构性失败：一参数曲线拟合、单参数模型及含49和241参数的多层感知机均收敛至同一伪真解，闭式匹配误差仅0.07%；而物理信息网络因复合目标函数，收敛至不同解。文中报告了该工具失效的预注册场景，即神经估计器在恢复性能上弱于Tikhonov正则化反演的情形，以及其保证成立但训练网络违反的前提假设。
原文摘要 · Abstract (English)
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. 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. 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. 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 $σ\geq 0.05,$ while the coefficient is wrong by $29.7\%$ at zero noise, $31.2\%$ at the loudest. 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. 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.
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