Paper Citation Record · LEDGER

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations

As of 11 August 2026, this Paper Citation Record lists 33 of 33 outbound references and 0 inbound Pith citation observations for arXiv:2507.12766.

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

pith.paper-citation-record.v1
2507.12766 v1

Coverage vector

measured 33 of 33 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T16:47:35.607059Z

measured 33 of 33 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

33 of 33 outbound references displayed

verified exact2
verified fuzzy8
unresolved23
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 7f57d284-4e75-4565-b305-cef22f79bd55 · outbound

This paper cites Yserentant, Sparse grid spaces for the numerical solu tion of the electronic schr¨ odinger equation, Numer.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Yserentant, Sparse grid spaces for the numerical solu tion of the electronic schr¨ odinger equation, Numer

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.829111Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 1abd070d-82cf-4a37-b773-fa03f8e12f74 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.823378Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation dda7ae2f-fd98-4fd9-ba7e-291f8ad05102 · outbound

This paper cites Ehrhardt, R.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Ehrhardt, R

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.816157Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation b5b96b26-128f-46b3-b93e-85a8cbcb3309 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.810821Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 79cd5782-0d36-448d-9042-97d66e3166fc · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 5

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.805018Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation eb9c51a0-e8d0-4629-a9e8-cba6f746faf2 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 6

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.799421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 87d09102-2d4a-4b64-b455-2c464cc39483 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.793615Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation eba60922-5c80-4f7b-b7d7-260fe96f2660 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.787447Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 49483fc7-ebde-4e35-997b-1badcca06903 · outbound

This paper cites Raissi, P.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Raissi, P

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.781705Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 59b83bd9-c397-487b-92ec-5ab2b0854372 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.776107Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 3c9f26d1-0bb6-48ee-ab85-8754f7da1a20 · outbound

This paper cites Yu, The deep Ritz method: a deep learning-based numer ical algorithm for solving variational problems, Commun.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Yu, The deep Ritz method: a deep learning-based numer ical algorithm for solving variational problems, Commun

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.770684Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 9f3e9da6-5b59-47cb-802a-8f7a9ad6bf09 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.765244Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 1b6c3862-abdb-4dac-b33e-98d8204d97ae · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 13

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.759627Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 6c4d2017-9626-4b96-aef3-debb847f4696 · outbound

This paper cites WANCO: Weak Adversarial Networks for Constrained Optimization problems.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations WANCO: Weak Adversarial Networks for Constrained Optimization problems

Reference 14

Resolution: verified exact; local_arxiv, observed 2026-08-06T16:47:35.655031Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation a53e7c6c-8139-4473-8062-27a7d877d2d8 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.754249Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation c7f27bee-c60e-49dc-b75c-2da3f20b51f2 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.748790Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation b673d044-82ed-4160-b2e6-83fd1220e418 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.742129Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 5ead70e5-c7d5-4d47-bbd9-8a90aadde714 · outbound

This paper cites R¨ ognvaldsson, On langevin updating in multilayer p erceptrons, Neural Comput.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations R¨ ognvaldsson, On langevin updating in multilayer p erceptrons, Neural Comput

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.736692Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 1ce1c984-688a-4517-b879-c911986df5ff · outbound

This paper cites Erhan, P.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Erhan, P

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.728452Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 3fc01543-9315-475f-8120-16d23be87cf7 · outbound

This paper cites Deep Learning Optimization Using Self-Adaptive Weighted Auxiliary Variables.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Deep Learning Optimization Using Self-Adaptive Weighted Auxiliary Variables

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-06T16:47:35.579415Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d6a3a215-408a-4c7c-b5d9-47c70670e08b · outbound

This paper cites Zhang, Z.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Zhang, Z

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.722404Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 963f1967-e379-4b25-aac6-62f74799145e · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.717184Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation a3f9c526-d2e0-488f-8a1b-75998cf80e24 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 23

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.710845Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 3a1cada3-2475-4069-9e02-d710ee9de824 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.705406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation ad18614d-5360-4c5e-be40-1a9e97f0b7bd · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.699448Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation d6eac249-9bf8-4c8b-a348-ee288e965055 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 26

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.693385Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 765d6343-4ed7-4a7c-a065-18b5cf480c92 · outbound

This paper cites Caragea, P.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Caragea, P

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T16:47:35.687229Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 104c9434-e699-459d-81b3-0194ec915ab1 · outbound

This paper cites Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't

Reference 28

Resolution: verified exact; local_arxiv, observed 2026-08-06T16:47:35.631304Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 98935d68-32c2-4553-be91-0dd84427edd1 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.680249Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation c033d079-bc85-4189-8aaa-c87fc5bf204d · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.673406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation f20f284a-c56c-48bb-90d6-8479594d9570 · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.667516Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 5fb45c22-e857-43ed-a4ca-b5df5123a63a · outbound

This paper cites an unresolved cited work.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-06T16:47:35.661269Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 762840bb-80ed-4dc9-96e6-867398ea3e0e · outbound

This paper cites Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory.

Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equations Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory

Reference 2020

Resolution: unresolved; no resolver link, observed 2026-08-06T16:47:35.571344Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.