2026-08-19 04:00 UTCOriginal source3 min readUpdated: 2026-08-19 08:49 UTC

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

arXiv:2608.16925v1 Announce Type: new 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. 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 $\sigma\geq 0.05,$ 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.

SourcearXiv Machine LearningAuthor: Eric Fock

-->

[Submitted on 4 Aug 2026]

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

View a PDF of the paper titled Detecting and Discriminating Operator Misspecification in Hybrid PDE-Parameter Learning: a Reference-Free Instrument, with Discrimination Bounded In Sample, by Eric Fock

View PDF HTML (experimental)

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. 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 $\sigma\geq 0.05,$ 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.

Comments: 14 pages, 8 figures. Supplementary material (5 pp.) included as an ancillary file

Subjects:

Machine Learning (cs.LG); Numerical Analysis (math.NA); Methodology (stat.ME)

MSC classes: 35R30, 62F03, 62F12, 65M32, 68T07

ACM classes: G.1.8; I.6.4; G.3; I.2.6

Cite as: arXiv:2608.16925 [cs.LG]

(or arXiv:2608.16925v1 [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)

Full-text links:

Access Paper:

View PDF

HTML (experimental)

TeX Source

view license

Ancillary-file links:

Ancillary files (details):

SUPPLEMENT.pdf

Current browse context:

cs.LG

new | recent | 2026-08

Change to browse by:

cs cs.NA math math.NA stat stat.ME

References & Citations

NASA ADS

Google Scholar

Semantic Scholar

Data provided by:

Bibliographic Tools

Bibliographic and Citation Tools

Bibliographic Explorer Toggle

Bibliographic Explorer (What is the Explorer?)

Connected Papers Toggle

Connected Papers (What is Connected Papers?)

Litmaps Toggle

Litmaps (What is Litmaps?)

scite.ai Toggle

scite Smart Citations (What are Smart Citations?)

Code, Data, Media

Code, Data and Media Associated with this Article

alphaXiv Toggle

alphaXiv (What is alphaXiv?)

Links to Code Toggle

CatalyzeX Code Finder for Papers (What is CatalyzeX?)

DagsHub Toggle

DagsHub (What is DagsHub?)

GotitPub Toggle

Gotit.pub (What is GotitPub?)

Huggingface Toggle

Hugging Face (What is Huggingface?)

ScienceCast Toggle

ScienceCast (What is ScienceCast?)

Demos

Replicate Toggle

Replicate (What is Replicate?)

Spaces Toggle

Hugging Face Spaces (What is Spaces?)

Spaces Toggle

TXYZ.AI (What is TXYZ.AI?)