发表机构
Université de La Réunion（留尼汪大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出一种无参考工具，可在混合PDE参数学习中仅通过一次拟合区分算子误设定与参数不可识别问题，实验验证其在样本内可有效判别两类失败情况。
AI 中文摘要
我们构建了一种工具，该工具仅需一次拟合且无需 oracle（先验知识），即可判断混合PDE参数估计器所假设的算子是否错误，并将其与仅无法识别的参数区分开。针对一个自伴抛物型逆问题，带有插件尺度和逐种子参数的信息矩阵统计量在正确设定下的中位数为0.19，对预注册上限0.10的拒绝率为0.033，在两种误设定下分别升至224和85，在所有重复中均触发。在正确设定但不可识别的设计下，该统计量保持静默——当n=200时为0.050，Clopper-Pearson区间为[0.024, 0.090]，而秩统计量在预注册边界c₅*=2.15×10⁻³处崩溃至0。因此，对一次拟合的两次读取即可在该工具适用的三种设计中区分两种失败情况，这种区分是核心贡献，仅检测则是竞争激烈的领域。该工具在样本内是有界的，在样本外是一种方向。之所以需要它，是因为常规精度检查存在盲区：误设定估计器的域内RMSE为2.7×10⁻²，在σ≥0.05时低于观测噪声，而系数在零噪声时错误29.7%，在最大噪声时错误31.2%。且这种失败并非架构性的：单参数曲线拟合、裸参数、参数为49和241的多层感知器收敛到相同的伪真值，匹配精度达0.07%，而物理信息神经网络因其复合目标收敛到不相交的伪真值。我们报告了该工具失效的情况、预注册的负面结果（神经估计器在恢复任务上输给Tikhonov正则化反演），以及其保证成立但训练网络违反的假设。
英文摘要
Physics-informed neural networks and hybrid models infer PDE coefficients from noisy data. When a trained network returns one, no standard check says whether to trust it. We show what those checks report when the operator is wrong: one sensor aggregating several diffusion sources. On one parabolic benchmark at $2\%$ noise, the in-domain error is $1.4$ times the noise while the identified diffusivity settles $30\%$ off. Every least-squares minimiser reaches that value, which drifts $27\%$ across windows; the network, whose objective is composite, settles $1.3\%$ away. The checks stay as silent when the design is blind to a rate of a richer operator, though the remedies are opposite. We develop a reference-free diagnostic, read in the physical parameter, not the weights, without retraining the network: an information-matrix test on the residuals, a heterogeneity statistic across window refits, and a Fisher-rank statistic on the design at the rates the single fit postulates. On the analytic head the specification test holds its pre-registered ceiling and rejects every misspecified replicate of both benchmark configurations, with a notch against a missing reaction term. The rank statistic is exactly zero only where the design is blind; a wrong operator confined to that mode leaves the specification test mute, and the rank statistic says so before any fit. The window reading exceeds its ceiling by one seed in thirty. A network frozen at its minimum returns the same verdicts; one stopped short rejects as a wrong operator would.
Comments12 pages, 6 figures, 7 tables. Supplementary material (15 pp.) included as an ancillary file
