arXiv：cs.LG· Eric Fock·· 3 小时前
物理信息系数学习的事后诊断：算子错误还是设计盲区？
Wrong Operator or Blind Design? A Post-Training Diagnostic for Physics-Informed Coefficient Learning
AI 导读
论文提出一种无需真值的训练后系数诊断方法，用三项经典统计检验追问训练模型：能否解释数据、设计能否分辨所命名的物理量、系数是否依赖窗口。在抛物基准上，正确算子可将系数恢复到机器精度，错误算子则偏差30%；7%噪声下预测误差低于噪声，系数仍偏差30%。规格检验保持预注册误报上限，拒绝所有被测错误算子重复样本，但设计盲区处检验会沉默。
正文
Abstract:Physics-informed and hybrid models integrate data and physics, and read learned coefficients as physical. Yet a small prediction error guarantees neither that the postulated operator is right, nor that the design identifies the coefficient, nor that training reached its minimum. On one parabolic benchmark where a sensor aggregates several diffusion rates and the truth is known, a correct operator recovers the coefficient to machine precision. A wrong one leaves it $30\%$ off while the in-domain error is 1.36 times the noise, and drifts $27\%$ across windows. At $7\%$ noise the error falls below the noise; the coefficient stays $30\%$ off. Without the truth, a post-training coefficient diagnostic questions the trained model with three classical statistics, without oracle; the first two need no refit. Does it explain the data? Can the design resolve the named physics? Does the coefficient depend on the window? The specification test keeps its pre-registered false-alarm ceiling and rejects every replicate of both wrong operators tested; a network frozen at its minimum returns the same verdicts. Where the design is blind, a wrong operator confined to the unseen rate leaves the test mute; the design reading flags it beforehand. A network stopped short of its minimum mimics a wrong operator; an analytic second fit separates the two. The test has a notch against a missing reaction term, miscalibrates under correlated noise and loses power beyond $7\%$ noise. Each warning names its remedy (operator, experiment or training); silence certifies nothing.
| Comments: | 13 pages, 5 figures, 6 tables. Supplementary material (20 pp.) included as an ancillary file |
| Subjects: | Machine Learning (cs.LG); Numerical Analysis (math.NA); Methodology (stat.ME) |
| MSC classes: | 35R30, 62F03, 62F12, 62K05, 65M32, 68T07 |
| ACM classes: | G.1.8; I.6.4; G.3; I.2.6 |
| Cite as: | arXiv:2608.16925 [cs.LG] |
| (or arXiv:2608.16925v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.16925 arXiv-issued DOI via DataCite |
Submission history
From: Eric Fock PhD [view email]
[v1]
Tue, 4 Aug 2026 21:20:44 UTC (562 KB)
[v2]
Wed, 30 Sep 2026 13:25:04 UTC (636 KB)
[v3]
Wed, 14 Oct 2026 09:40:45 UTC (734 KB)
来源：arXiv：cs.LG · arxiv.org