Paper Citation Record · LEDGER

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability

As of 8 August 2026, this Paper Citation Record lists 31 of 31 outbound references and 0 inbound Pith citation observations for arXiv:2506.20181.

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

pith.paper-citation-record.v1
2506.20181 v2

Coverage vector

measured 31 of 31 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T23:01:58.650067Z

measured 31 of 31 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+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

31 of 31 outbound references displayed

verified exact0
verified fuzzy26
unresolved3
parse uncertain0
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metadata mismatch2

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation e1505d47-d2a6-47da-b0a9-abee9154f0f8 · outbound

This paper cites The Mathematical Theory of Finite Element Methods.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability The Mathematical Theory of Finite Element Methods

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.959537Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c2680406-369e-4491-99ec-0da7d7e17b83 · outbound

This paper cites Chebyshev and Fourier Spectral Methods.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Chebyshev and Fourier Spectral Methods

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.950680Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b3a3b06e-9e67-4662-89ef-85d39364ed05 · outbound

This paper cites Runge–kutta discontinuous galerkin methods for convection-dominated problems.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Runge–kutta discontinuous galerkin methods for convection-dominated problems

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.941918Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 972961fc-5188-49f5-97e8-4965e4401e41 · outbound

This paper cites A Multigrid Tutorial.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability A Multigrid Tutorial

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.932681Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 5de68c1f-9e37-490c-83cb-d06c4913a38d · outbound

This paper cites Convergence and error analysis of physics-informed neural networks.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Convergence and error analysis of physics-informed neural networks

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.924194Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 642bf1d3-f273-49df-bda8-e88831146c05 · outbound

This paper cites Numerical Solution of Time-dependent Advection- diffusion-reaction Equations.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Numerical Solution of Time-dependent Advection- diffusion-reaction Equations

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.914445Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 5a78be35-d75b-48f0-a392-ddb85e8d52ea · outbound

This paper cites A matrix-free finite element method for solid mechanics.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability A matrix-free finite element method for solid mechanics

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.905801Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation cd0217da-054c-4f68-b3a4-517757ab6e1e · outbound

This paper cites Automated Solution of Differential Equa- tions by the Finite Element Method.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Automated Solution of Differential Equa- tions by the Finite Element Method

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.896667Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c1297aa3-7044-4036-aae0-819395e05276 · outbound

This paper cites Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear pdes.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear pdes

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.887625Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c1270301-af24-4039-a332-57c39b73329b · outbound

This paper cites Fourier neural operator for parametric partial differential equations.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Fourier neural operator for parametric partial differential equations

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.878753Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 25e9aa45-e066-4919-be81-88af73e550f1 · outbound

This paper cites Neural operators: Learning maps between function spaces.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Neural operators: Learning maps between function spaces

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.869972Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 35498afa-bfda-4500-a8ec-55a73c4a4576 · outbound

This paper cites Learning nonlinear operators via deep- onet based on the universal approximation theorem of operators.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Learning nonlinear operators via deep- onet based on the universal approximation theorem of operators

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.861712Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 5b737718-7d2c-4a67-8ecb-dc0837b77248 · outbound

This paper cites Marrying causal representation learning with dynamical systems.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Marrying causal representation learning with dynamical systems

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.853241Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation ab656d33-0c40-4541-8a1a-388f6cd000e1 · outbound

This paper cites Robust discovery of partial differential equations in complex situations.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Robust discovery of partial differential equations in complex situations

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.844307Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 4ff59a9f-cd1d-40bb-98b1-132a5039c60f · outbound

This paper cites A causal graph ode for continuous-time treatment effect modeling.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability A causal graph ode for continuous-time treatment effect modeling

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.835911Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 998e8acd-22f4-49eb-baf6-7027f7d1120e · outbound

This paper cites Discovering governing equations from data by sparse identification of nonlinear dynamical systems.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Discovering governing equations from data by sparse identification of nonlinear dynamical systems

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.827484Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b4e1c0f4-bec9-42de-832f-4bebfbedcff2 · outbound

This paper cites Data-driven dis- covery of partial differential equations.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Data-driven dis- covery of partial differential equations

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.818317Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1fa930db-5451-4dde-8074-ce97296eb020 · outbound

This paper cites Detecting Entrainment in Fermi-Bose Mixtures.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Detecting Entrainment in Fermi-Bose Mixtures

Reference 18

Resolution: metadata mismatch; local_arxiv, observed 2026-08-06T23:01:58.716083Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 8c43fc44-601c-4e79-bc82-a781b0dcb9ce · outbound

This paper cites Ensemble learning for robust pde discovery.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Ensemble learning for robust pde discovery

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.809917Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation d10556c3-1dca-45d5-baa6-b72e3ffed4bf · outbound

This paper cites Learning deep koopman representations for data-efficient control.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Learning deep koopman representations for data-efficient control

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.801159Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation a6a46f4a-f2be-4e83-a639-5811e4ffe7ba · outbound

This paper cites Hybrid ² neural ode causal modeling.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Hybrid ² neural ode causal modeling

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.792529Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 2729c5cb-54af-43e8-bf2e-285182d9e7da · outbound

This paper cites Impact of QCD sum rules coupling constants on neutron stars structure.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Impact of QCD sum rules coupling constants on neutron stars structure

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-06T23:01:58.623805Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bb761284-714f-4aaf-a602-312190daf267 · outbound

This paper cites Vacuum radiation versus shortcuts to adiabaticity.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Vacuum radiation versus shortcuts to adiabaticity

Reference 23

Resolution: metadata mismatch; local_arxiv, observed 2026-08-06T23:01:58.693483Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3e50b51d-034e-4ef3-a72d-f7fa170d4a6b · outbound

This paper cites Causality: Models, Reasoning, and Inference.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Causality: Models, Reasoning, and Inference

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.783421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 65090253-0e98-461f-a19b-a3896dec0e46 · outbound

This paper cites Toward causal representation learning.PNAS, 118(38):e2103171118, 2021.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Toward causal representation learning.PNAS, 118(38):e2103171118, 2021

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.774404Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation a86a56c7-034d-4b7c-b9a9-a46dafb86e5d · outbound

This paper cites On the spectral radius of the non-backtracking matrix of the configuration model.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability On the spectral radius of the non-backtracking matrix of the configuration model

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-06T23:01:58.635561Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ee1109fa-ced7-4157-a03a-5b4329a476cd · outbound

This paper cites Is l2 loss always suitable for pinn training? In Advances in Neural Information Processing Systems 35 , pages 4021–4033, 2022.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Is l2 loss always suitable for pinn training? In Advances in Neural Information Processing Systems 35 , pages 4021–4033, 2022

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.764713Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 0f47c9e8-f92e-4cd6-88de-829954a993ce · outbound

This paper cites Physpde: A physical hypothesis selection benchmark for pde discovery.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Physpde: A physical hypothesis selection benchmark for pde discovery

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.754015Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 04c2d4d0-b84a-41ee-bcab-9ea580271d00 · outbound

This paper cites Cand` es, Justin Romberg, and Terence Tao.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Cand` es, Justin Romberg, and Terence Tao

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.744525Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 53c1caea-f375-429e-b2c5-d04db3e6c9c1 · outbound

This paper cites an unresolved cited work.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-06T23:01:58.734984Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation fc6939db-a41d-400f-8770-064ac58e079e · outbound

This paper cites A Mathematical Introduction to Compressive Sensing.

Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability A Mathematical Introduction to Compressive Sensing

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T23:01:58.725533Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Pith citing papers

No inbound Pith citation observations are available.