Paper Citation Record · LEDGER

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

As of 19 August 2026, this Paper Citation Record lists 28 of 28 outbound references and 0 inbound Pith citation observations for arXiv:2606.12050.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2606.12050 v1

Coverage vector

measured 28 of 28 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-06-27T10:56:10.330401Z

measured 28 of 28 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-19T06:32:44.657259+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

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measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

28 of 28 outbound references displayed

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verified fuzzy0
unresolved10
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External citation measurements

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Outbound references

Observation d1e5023d-f204-4f06-b27c-2c72d51d4106 · outbound

This paper cites Hillebrecht, B.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Hillebrecht, B

Reference 1

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b90bb7b2-8274-4e8c-b089-a48ae256eb61 · outbound

This paper cites Cuomo, et al., Scientific machine learning through physics-informed neural networks: where we are and what’s next, Journal of Scientific Computing 92 (2022) 88.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Cuomo, et al., Scientific machine learning through physics-informed neural networks: where we are and what’s next, Journal of Scientific Computing 92 (2022) 88

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

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Observation 3e3a2428-4252-4738-98c3-d9843f950ca1 · outbound

This paper cites Raissi, P.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Raissi, P

Reference 3

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.513850Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation e065a212-5b72-4189-8604-f18aa3248fa3 · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 4

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d09adc2f-bebc-4a27-8a44-b7d960999fe2 · outbound

This paper cites Cortés-Ciriano, A.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Cortés-Ciriano, A

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4ad07fd2-da56-42bd-ab77-0dd14bdfb65e · outbound

This paper cites A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem

Reference 6

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.143091Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 9778d128-9e28-46a3-a37b-d155aadf909b · outbound

This paper cites Optimal parameters related with continuity properties of the multilinear fractional integral operator between Lebesgue and Lipschitz spaces.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Optimal parameters related with continuity properties of the multilinear fractional integral operator between Lebesgue and Lipschitz spaces

Reference 7

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.131826Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation b044dd8f-8e1d-48aa-89b2-49208589093e · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 067eeef9-f845-487c-9fec-24cd1f0622db · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 9

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

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Observation f63ecaa6-4f82-4456-b9fe-e4dca384b4be · outbound

This paper cites Generic bounds on the approximation error for physics-informed (and) operator learning.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Generic bounds on the approximation error for physics-informed (and) operator learning

Reference 10

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.140161Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation f2deada7-3cf1-4868-ba25-91278f0e3c58 · outbound

This paper cites Haugen, A.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Haugen, A

Reference 11

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.126161Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation a73b9dd3-ae7d-408a-8089-af9068a74a84 · outbound

This paper cites Hillebrecht, B.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Hillebrecht, B

Reference 12

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d137cf87-d1a8-4414-8d1a-6300e011de2d · outbound

This paper cites Prediction error certification for PINNs: Theory, computation, and application to Stokes flow.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

Reference 13

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.137434Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 08636eaa-5c86-4161-8856-c82abf477130 · outbound

This paper cites Eiras, A.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Eiras, A

Reference 14

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bd776574-5326-47eb-bbe7-7b185ad4b62e · outbound

This paper cites Neural functional a posteriori error estimates.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Neural functional a posteriori error estimates

Reference 15

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.128846Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 27255a98-d619-4b06-808d-4fccf7330f9c · outbound

This paper cites Energy-based error bound of physics-informed neural network solutions in elasticity.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Energy-based error bound of physics-informed neural network solutions in elasticity

Reference 16

Resolution: verified exact; arxiv_id, observed 2026-07-03T08:17:45.134391Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 2b6c71a4-74e6-4b0a-a7e7-fafad5d92297 · outbound

This paper cites Chowell, L.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Chowell, L

Reference 17

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.502339Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 2ab7c5ed-71ea-4c8b-a9df-798e7bd41949 · outbound

This paper cites Diekmann, J.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Diekmann, J

Reference 18

Resolution: malformed identifier; arxiv_id, observed 2026-07-03T08:17:45.123382Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 0fa626af-2855-4845-a37b-7636339b0251 · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 19

Resolution: malformed identifier; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e418c54f-22b7-4420-afe1-f56ff8b5c4af · outbound

This paper cites Schuster, What is special about autocatalysis?, Monat- shefte für Chemie – Chemical Monthly (2019).doi:10.1007/ s00706-019-02437-z.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Schuster, What is special about autocatalysis?, Monat- shefte für Chemie – Chemical Monthly (2019).doi:10.1007/ s00706-019-02437-z

Reference 20

Resolution: malformed identifier; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4bd7ea06-5347-445e-b0fc-a09c6083eeaa · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 21

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.503913Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 18c70285-307c-4de2-9fe9-fbcabdd4aea1 · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 22

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.499540Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 71ea7aac-988f-48c3-bd63-17a6f3f8db1d · outbound

This paper cites SIAM Journal on Scientific Computing , keywords =.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds SIAM Journal on Scientific Computing , keywords =

Reference 23

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.509681Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 5b4ca24a-7110-4a66-a8d8-63d8bc49f69a · outbound

This paper cites Stability in Training PINNs for Stiff PDEs: Why Initial Conditions Matter.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Stability in Training PINNs for Stiff PDEs: Why Initial Conditions Matter

Reference 24

Resolution: verified exact; arxiv_id, observed 2026-08-18T02:13:04.572550Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation f64530a5-7237-46fd-8f15-8a0823f5d23b · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 25

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1c68d000-8e82-46d4-830a-02ba9d8fd654 · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 26

Resolution: unresolved; no resolver link, observed 2026-06-27T10:56:10.330401Z

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Unavailable: canonical work link unavailable.

Observation 1b96b2ed-1ba6-4b63-b6ca-ef89e0062b13 · outbound

This paper cites and Ratiu, Tudor S.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds and Ratiu, Tudor S

Reference 27

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.501957Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 8c5c81c6-cba6-48c0-a44b-3692b76c9fa4 · outbound

This paper cites an unresolved cited work.

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds Unresolved cited work

Reference 28

Resolution: verified exact; doi, observed 2026-06-27T11:00:50.511672Z

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No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Pith citing papers

No inbound Pith citation observations are available.