Paper Citation Record · LEDGER

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media

As of 9 August 2026, this Paper Citation Record lists 33 of 33 outbound references and 0 inbound Pith citation observations for arXiv:2606.25820.

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

pith.paper-citation-record.v1
2606.25820 v1

Coverage vector

measured 33 of 33 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-06-25T20:04:35.669107Z

measured 33 of 33 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

33 of 33 outbound references displayed

verified exact5
verified fuzzy0
unresolved26
parse uncertain0
malformed identifier1
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation e133a546-9ed9-4a50-8cc9-0fbafd158719 · outbound

This paper cites Nonlinear model order reduction based on local reduced-order bases.International Journal for Numerical Methods in Engineering, 92(10):891–916, 2012.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Nonlinear model order reduction based on local reduced-order bases.International Journal for Numerical Methods in Engineering, 92(10):891–916, 2012

Reference 1

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6de0e666-7e9e-4e03-af5a-d69799750194 · outbound

This paper cites Geometric multi- grid methods for darcy–forchheimer flow in fractured porous media.Computers & Mathematics with Applications, 78(9):3139–3151, 2019.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Geometric multi- grid methods for darcy–forchheimer flow in fractured porous media.Computers & Mathematics with Applications, 78(9):3139–3151, 2019

Reference 2

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

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Observation e2b7acdc-09dc-4c23-86ee-fe640df2b4a2 · outbound

This paper cites Analysis of generalized forchheimer flows of compressible fluids in porous media.Journal of Mathematical Physics, 50(10), 2009.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Analysis of generalized forchheimer flows of compressible fluids in porous media.Journal of Mathematical Physics, 50(10), 2009

Reference 3

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ae77fb88-b9e5-48f1-97a3-d79ce8a40cdb · outbound

This paper cites SIAM, 2017.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media SIAM, 2017

Reference 4

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

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Observation 76d5a1f6-c141-46e4-89f4-cf5aa4989b1c · outbound

This paper cites An introduction to the proper orthogonal decomposition.Current science, pages 808–817, 2000.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media An introduction to the proper orthogonal decomposition.Current science, pages 808–817, 2000

Reference 5

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f8792a5a-0c14-4f7e-82e1-2f683ba2a7c8 · outbound

This paper cites Discrete empirical interpolation for nonlinear model reduction.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Discrete empirical interpolation for nonlinear model reduction

Reference 6

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d27d6c98-9098-46b9-aaed-0cbfeec23831 · outbound

This paper cites An adaptive gmsfem for high-contrast flow problems.Journal of Computational Physics, 273:54–76, 2014.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media An adaptive gmsfem for high-contrast flow problems.Journal of Computational Physics, 273:54–76, 2014

Reference 7

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f2e4faae-2717-4c4e-88df-51478439bcea · outbound

This paper cites Generalized multiscale finite element methods.Journal of Computational Physics, 251:116–135, 2013.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Generalized multiscale finite element methods.Journal of Computational Physics, 251:116–135, 2013

Reference 8

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation eb45cc17-beda-46ce-9682-885ae3535a67 · outbound

This paper cites Springer, 2009.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Springer, 2009

Reference 9

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b5677d63-d785-4329-8de4-d7e80375cd4c · outbound

This paper cites Spectral neural operators.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Spectral neural operators

Reference 10

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

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Observation 7e1f4227-149a-4c53-912d-b3352e005e87 · outbound

This paper cites Localized model order reduction in porous media flow simulation.Journal of Petroleum Science and Engineering, 145:689–703, 2016.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Localized model order reduction in porous media flow simulation.Journal of Petroleum Science and Engineering, 145:689–703, 2016

Reference 11

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 10e6e968-d02c-403d-aae4-9ac12f23b368 · outbound

This paper cites Model order reduction in porous media flow simulation using quadratic bilinear formulation.Computational Geosciences, 20(3):723–735, 2016.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Model order reduction in porous media flow simulation using quadratic bilinear formulation.Computational Geosciences, 20(3):723–735, 2016

Reference 12

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b6172e13-6b64-4fef-b026-886e1ac5a4e9 · outbound

This paper cites Springer, 2016.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Springer, 2016

Reference 13

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a50f9188-b778-41cb-a117-638ff6ea8dff · outbound

This paper cites Operator learning: Algorithms and analysis.Handbook of Numerical Analysis, 25:419–467, 2024.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Operator learning: Algorithms and analysis.Handbook of Numerical Analysis, 25:419–467, 2024

Reference 14

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fcb48c97-e066-4eea-a257-13da8dea6c53 · outbound

This paper cites Fourier Neural Operator for Parametric Partial Differential Equations.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Fourier Neural Operator for Parametric Partial Differential Equations

Reference 15

Resolution: verified exact; local_arxiv, observed 2026-07-04T20:30:08.134863Z

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 8eaee94d-b534-4e53-b391-98a02d101332 · outbound

This paper cites Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

Reference 16

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bb62f0c1-9205-46b0-8ff0-21e8c580c839 · outbound

This paper cites A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022

Reference 17

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1d9d19f7-df29-448b-be40-1ee46779c6fd · outbound

This paper cites Neural-pod: A plug-and-play neural opera- tor framework for infinite-dimensional functional nonlinear proper orthogonal decomposition.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Neural-pod: A plug-and-play neural opera- tor framework for infinite-dimensional functional nonlinear proper orthogonal decomposition

Reference 18

Resolution: verified exact; arxiv_id, observed 2026-07-04T20:30:08.131244Z

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 7a9ebade-f47c-459f-b132-d1d1a3bed5dd · outbound

This paper cites Learning two-phase microstructure evolution using neural operators and autoencoder architectures.npj Computational Materials, 8(1):190, 2022.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Learning two-phase microstructure evolution using neural operators and autoencoder architectures.npj Computational Materials, 8(1):190, 2022

Reference 19

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9b74af76-8d4c-43d0-b14f-7786d4d73906 · outbound

This paper cites Pytorch: An impera- tive style, high-performance deep learning library.Advances in neural information processing systems, 32, 2019.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Pytorch: An impera- tive style, high-performance deep learning library.Advances in neural information processing systems, 32, 2019

Reference 20

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d1fbdb24-f2b7-4e56-8447-da78ee7fa034 · outbound

This paper cites Springer, 2015.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Springer, 2015

Reference 21

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 66abc754-0c43-42e9-9b26-8126fbd53991 · outbound

This paper cites Certified reduced basis approximation for parametrized partial differential equations and applications.Journal of Mathematics in Industry, 1(1):3, 2011.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Certified reduced basis approximation for parametrized partial differential equations and applications.Journal of Mathematics in Industry, 1(1):3, 2011

Reference 22

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4ea9983e-4a94-4edb-8083-645253bae860 · outbound

This paper cites Locally subspace-informed neural operators for efficient multiscale pde solving.arXiv preprint arXiv:2505.16030, 2025.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Locally subspace-informed neural operators for efficient multiscale pde solving.arXiv preprint arXiv:2505.16030, 2025

Reference 23

Resolution: verified exact; arxiv_id, observed 2026-07-04T20:30:08.138121Z

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 0a331db6-6094-49c2-a7f5-80a7253cf189 · outbound

This paper cites Ensemble and Mixture-of-Experts DeepONets For Operator Learning.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Ensemble and Mixture-of-Experts DeepONets For Operator Learning

Reference 24

Resolution: verified exact; arxiv_id, observed 2026-07-04T20:30:08.141386Z

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 346d8283-146f-443f-99e9-2d5eb201645d · outbound

This paper cites Wavelet neural operator: a neural operator for parametric partial differential equations.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Wavelet neural operator: a neural operator for parametric partial differential equations

Reference 25

Resolution: metadata mismatch; arxiv_id, observed 2026-07-04T20:30:08.144530Z

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 64d14c0b-920b-44ee-8710-e6005d551cee · outbound

This paper cites an unresolved cited work.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Unresolved cited work

Reference 26

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 56a5819a-08f8-4d53-b234-bd09a4365c86 · outbound

This paper cites Learn- ing macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques.Journal of Computational Physics, 412:109323, 2020.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Learn- ing macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques.Journal of Computational Physics, 412:109323, 2020

Reference 27

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bd60d095-531a-43c2-a814-64435db35f8c · outbound

This paper cites an unresolved cited work.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Unresolved cited work

Reference 28

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 00675c44-1176-4f43-871f-ff898aef379c · outbound

This paper cites an unresolved cited work.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Unresolved cited work

Reference 29

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 25edd021-3915-433b-a0c4-257426389094 · outbound

This paper cites Machine learning for accelerating macroscopic parame- ters prediction for poroelasticity problem in stochastic media.Computers & Mathematics with Applications, 84:185–202, 2021.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Machine learning for accelerating macroscopic parame- ters prediction for poroelasticity problem in stochastic media.Computers & Mathematics with Applications, 84:185–202, 2021

Reference 30

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 53118afa-d694-4af5-88af-b2fa9e3cb1d8 · outbound

This paper cites Scipy 1.0: fundamental algorithms for scientific computing in python.Nature methods, 17(3):261– 272, 2020.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Scipy 1.0: fundamental algorithms for scientific computing in python.Nature methods, 17(3):261– 272, 2020

Reference 31

Resolution: unresolved; no resolver link, observed 2026-06-25T20:04:35.669107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 26d2673c-95b5-4680-8cbb-6dd55edae386 · outbound

This paper cites Reduced-basis deep operator learning for parametric pdes with independently varying boundary and source data.arXiv preprint arXiv:2511.18260, 2025.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Reduced-basis deep operator learning for parametric pdes with independently varying boundary and source data.arXiv preprint arXiv:2511.18260, 2025

Reference 32

Resolution: verified exact; arxiv_id, observed 2026-07-04T20:30:08.147596Z

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 c543e7ae-38f1-4ba9-a5ee-f7477fbd00a5 · outbound

This paper cites Finite element representation network (fern) for operator learning with a localized trainable basis.arXiv preprint arXiv:2510.26962, 2025.

Operator Learning on the Data-Driven Multiscale Space for Nonlinear Flow in Random Heterogeneous Porous Media Finite element representation network (fern) for operator learning with a localized trainable basis.arXiv preprint arXiv:2510.26962, 2025

Reference 33

Resolution: malformed identifier; arxiv_id, observed 2026-07-04T20:30:08.151309Z

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.