Paper Citation Record · LEDGER

Recurrent Neural Operators: Stable Long-Term PDE Prediction

As of 9 August 2026, this Paper Citation Record lists 33 of 33 outbound references and 6 inbound Pith citation observations for arXiv:2505.20721.

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

pith.paper-citation-record.v1
2505.20721 v1

Coverage vector

measured 33 of 33 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T13:53:38.626157Z

measured 39 of 39 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-09T06:31:02.800959+00:00

measured 6 of 6 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-03T13:08:22.792062Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-06-29T22:24:00.563121Z

Reference resolution

33 of 33 outbound references displayed

verified exact0
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External citation measurements

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Outbound references

Observation 1ac83e79-b77d-4644-990c-ba49cdb3a5ec · outbound

This paper cites LNO: Laplace Neural Operator for Solving Differential Equations.Nature Machine Intelligence, 6(6):631–640, June 2024.

Recurrent Neural Operators: Stable Long-Term PDE Prediction LNO: Laplace Neural Operator for Solving Differential Equations.Nature Machine Intelligence, 6(6):631–640, June 2024

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:47.197429Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 94eeab2a-bb0f-4159-8619-840bcf73ee9b · outbound

This paper cites an unresolved cited work.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-07T13:53:46.963318Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation a7947323-8fa2-4e56-ac16-6a30a23be19e · outbound

This paper cites The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems.Communications in Mathematics and Statistics, 6(1):1–12, February 2018.

Recurrent Neural Operators: Stable Long-Term PDE Prediction The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems.Communications in Mathematics and Statistics, 6(1):1–12, February 2018

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:46.564074Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation c9fe57f8-6240-4110-8ab2-8b1456caf5a6 · outbound

This paper cites MgNO: Efficient Parameterization of Linear Operators via Multigrid.

Recurrent Neural Operators: Stable Long-Term PDE Prediction MgNO: Efficient Parameterization of Linear Operators via Multigrid

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:46.333645Z

Source-reported events for the cited work

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Observation ea85af77-d80a-4e4a-b720-8080bd806a69 · outbound

This paper cites State-space models are accurate and efficient neural operators for dynamical systems.

Recurrent Neural Operators: Stable Long-Term PDE Prediction State-space models are accurate and efficient neural operators for dynamical systems

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-07T13:53:34.567646Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5611f74f-5e4b-404c-9646-8b6d75597f77 · outbound

This paper cites Deep backward schemes for high-dimensional nonlinear PDEs.Mathematics of Computation, 89(324):1547–1579, January 2020.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Deep backward schemes for high-dimensional nonlinear PDEs.Mathematics of Computation, 89(324):1547–1579, January 2020

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:46.045023Z

Source-reported events for the cited work

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Observation 6d6f7d96-ebed-4faf-bbff-1b8ed3edff33 · outbound

This paper cites DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation.

Recurrent Neural Operators: Stable Long-Term PDE Prediction DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:45.732731Z

Source-reported events for the cited work

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Observation 56e81580-b267-47d1-ab5b-b8ae8ab8622c · outbound

This paper cites Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws.Communications in Computational Physics, 37(2):420–456, January 2025.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws.Communications in Computational Physics, 37(2):420–456, January 2025

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:45.401258Z

Source-reported events for the cited work

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Observation e2700e91-69c7-470b-8e8f-b279712db830 · outbound

This paper cites On Universal Approximation and Error Bounds for Fourier Neural Operators.Journal of Machine Learning Research, 22(290):1–76, January 2021.

Recurrent Neural Operators: Stable Long-Term PDE Prediction On Universal Approximation and Error Bounds for Fourier Neural Operators.Journal of Machine Learning Research, 22(290):1–76, January 2021

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:45.058524Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation b9ce907b-1892-4082-abe4-c2d871cb926e · outbound

This paper cites Neural Operator: Learning Maps Between Function Spaces With Applica- tions to PDEs.Journal of Machine Learning Research, 24(89):1–97, January 2023.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Neural Operator: Learning Maps Between Function Spaces With Applica- tions to PDEs.Journal of Machine Learning Research, 24(89):1–97, January 2023

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:44.747823Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 13190015-7f7b-4a50-a870-265edf390f08 · outbound

This paper cites Fourier Neural Operator with Learned Deformations for PDEs on General Geometries.The Journal of Machine Learning Research, 24(1):1532–4435, January 2023.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Fourier Neural Operator with Learned Deformations for PDEs on General Geometries.The Journal of Machine Learning Research, 24(1):1532–4435, January 2023

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:44.368692Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 8d424cde-e803-4d20-983f-ee6f08b5e8f6 · outbound

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

Recurrent Neural Operators: Stable Long-Term PDE Prediction Fourier Neural Operator for Parametric Partial Differential Equations

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:44.128431Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation e9455b73-b951-43f7-bc34-ad6d88486970 · outbound

This paper cites Geometry-Informed Neural Operator for Large-Scale 3D PDEs.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Geometry-Informed Neural Operator for Large-Scale 3D PDEs

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:43.868492Z

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Observation 521c144d-1c5f-42f0-acee-5e16dd013680 · outbound

This paper cites Veeling, Paris Perdikaris, Richard E.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Veeling, Paris Perdikaris, Richard E

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:43.713402Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation ae7a3573-7b0d-4d3f-b539-512fc161f87e · outbound

This paper cites Mitigating spectral bias for the multiscale operator learning.Journal of Computational Physics, 506:112944, June 2024.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Mitigating spectral bias for the multiscale operator learning.Journal of Computational Physics, 506:112944, June 2024

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:43.404374Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation eedc4ecb-a835-4394-96a2-fc9b92522eea · outbound

This paper cites A MgNO Method for Multiphase Flow in Porous Media.

Recurrent Neural Operators: Stable Long-Term PDE Prediction A MgNO Method for Multiphase Flow in Porous Media

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:43.037295Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 45c3e1c9-d965-4718-962f-8aa051e1c706 · outbound

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

Recurrent Neural Operators: Stable Long-Term PDE Prediction Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:42.823448Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 26ca6241-b5b1-42c6-ac25-3b4f4530bd3f · outbound

This paper cites Towards Stability of Autoregressive Neural Operators.Transactions on machine learning research, October 2023.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Towards Stability of Autoregressive Neural Operators.Transactions on machine learning research, October 2023

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:42.344758Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation d69671b7-92fd-412e-b251-80229aee77c5 · outbound

This paper cites Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:41.985862Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 85cbf4cc-824d-49f1-8517-d998b714a030 · outbound

This paper cites FourCastNet: A Global Data- driven High-resolution Weather Model using Adaptive Fourier Neural Operators.

Recurrent Neural Operators: Stable Long-Term PDE Prediction FourCastNet: A Global Data- driven High-resolution Weather Model using Adaptive Fourier Neural Operators

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:41.725340Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 3c91a942-c6d2-4231-8a2e-e112d917fa72 · outbound

This paper cites Forward–Backward Stochastic Neural Networks: Deep Learning of High- Dimensional Partial Differential Equations.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Forward–Backward Stochastic Neural Networks: Deep Learning of High- Dimensional Partial Differential Equations

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:41.436522Z

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Observation 2d3d53cc-9e26-4759-8913-e4480d3a3aec · outbound

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Recurrent Neural Operators: Stable Long-Term PDE Prediction Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-07T13:53:41.182124Z

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No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 0499367d-969f-4fff-b384-435b894df9cc · outbound

This paper cites Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs

Reference 23

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Observation 9e2a60fd-eea1-470e-86b9-e810e9929f65 · outbound

This paper cites DGM: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, December 2018.

Recurrent Neural Operators: Stable Long-Term PDE Prediction DGM: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, December 2018

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:40.912840Z

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Observation ddf73d44-ddb4-436d-8b87-e28a5231c454 · outbound

This paper cites Factorized Fourier Neural Operators.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Factorized Fourier Neural Operators

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:40.729416Z

Source-reported events for the cited work

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Observation 9bbb95ea-781e-49dc-816f-8c557dfc7618 · outbound

This paper cites COAST: Intelligent Time-Adaptive Neural Operators.

Recurrent Neural Operators: Stable Long-Term PDE Prediction COAST: Intelligent Time-Adaptive Neural Operators

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-07T13:53:37.611152Z

Source-reported events for the cited work

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Observation 55108148-b9b0-4632-bdcd-f1f848fc215a · outbound

This paper cites Amortized Fourier Neural Operators.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Amortized Fourier Neural Operators

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:40.540572Z

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Observation 9250f5aa-c834-467a-b6da-754e31b27bcb · outbound

This paper cites Transfer Learning Enhanced DeepONet for Long- Time Prediction of Evolution Equations.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Transfer Learning Enhanced DeepONet for Long- Time Prediction of Evolution Equations

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:40.293111Z

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Observation b4da9aa1-3887-4b2c-b749-acfad9b9e2bf · outbound

This paper cites FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations.

Recurrent Neural Operators: Stable Long-Term PDE Prediction FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations

Reference 29

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Observation 38192298-6f44-4863-8976-397c0a970c50 · outbound

This paper cites Reliable extrapolation of deep neural operators informed by physics or sparse observations.Computer Methods in Applied Mechanics and Engineering, 412:116064, July 2023.

Recurrent Neural Operators: Stable Long-Term PDE Prediction Reliable extrapolation of deep neural operators informed by physics or sparse observations.Computer Methods in Applied Mechanics and Engineering, 412:116064, July 2023

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T13:53:40.035255Z

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Observation b3810c2f-786c-499d-84fe-9d06d5478ccd · outbound

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Recurrent Neural Operators: Stable Long-Term PDE Prediction Unresolved cited work

Reference 31

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Reference 32

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Observation 877911d3-872d-486c-be32-3a6a6b081797 · outbound

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Recurrent Neural Operators: Stable Long-Term PDE Prediction Unresolved cited work

Reference 33

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Pith citing papers

Observation cc18e0e8-d9b0-4e61-897d-3b52bc9f393b · inbound

Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling cites this paper.

Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 46

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Observation 14ee7b02-aca2-42ca-9180-4fccb7ad5210 · inbound

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators cites this paper.

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 11

Resolution: metadata mismatch; arxiv_id, observed 2026-05-12T03:26:19.757874Z

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Observation 35af1503-2eaa-475c-9676-4639f113a48f · inbound

Autoregression-Free Neural Operators for Time-Dependent PDEs cites this paper.

Autoregression-Free Neural Operators for Time-Dependent PDEs Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 35

Resolution: metadata mismatch; arxiv_id, observed 2026-06-29T22:24:00.564858Z

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Observation 479298eb-f1ca-4ff5-99d6-47005a3bda1f · inbound

Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses cites this paper.

Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 35

Resolution: unresolved; no resolver link, observed 2026-08-02T09:07:17.976701Z

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Observation 3c6eed9b-2d1c-47d0-9f2f-5c504fa6a5ba · inbound

No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study cites this paper.

No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 34

Resolution: unresolved; no resolver link, observed 2026-07-30T16:23:26.545464Z

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Observation fe3f1fef-6347-466a-b4f0-bdbe9cbfd864 · inbound

HERO: History-Enriched Rollout Training for Long-Horizon Autoregressive Neural Operators cites this paper.

HERO: History-Enriched Rollout Training for Long-Horizon Autoregressive Neural Operators Recurrent Neural Operators: Stable Long-Term PDE Prediction

Reference 51

Resolution: unresolved; no resolver link, observed 2026-08-03T13:08:22.792062Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.