Paper Citation Record · LEDGER

Scale-Consistent Learning for Partial Differential Equations

As of 19 August 2026, this Paper Citation Record lists 47 of 47 outbound references and 0 inbound Pith citation observations for arXiv:2507.18813.

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

pith.paper-citation-record.v1
2507.18813 v1

Coverage vector

measured 47 of 47 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-15T18:15:42.502816Z

measured 47 of 47 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.

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

47 of 47 outbound references displayed

verified exact2
verified fuzzy15
unresolved30
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b9ffa0f7-3588-404e-82de-6ccf0aa618d5 · outbound

This paper cites Neural operators for accelerating scientific simulations and design.Nature Reviews Physics, pages 1–9, 2024.

Scale-Consistent Learning for Partial Differential Equations Neural operators for accelerating scientific simulations and design.Nature Reviews Physics, pages 1–9, 2024

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.662121Z

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 ca1d45f5-7dd8-4a82-a573-5a663768ec84 · outbound

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

Scale-Consistent Learning for Partial Differential Equations FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.213149Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f7728f3b-58de-400d-8a47-278dd3d3a6aa · outbound

This paper cites Fourier Neural Operator for Plasma Modelling.

Scale-Consistent Learning for Partial Differential Equations Fourier Neural Operator for Plasma Modelling

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.221359Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cc10e224-0574-4f02-ba21-dd144346f4e6 · outbound

This paper cites Neural Operator: Graph Kernel Network for Partial Differential Equations.

Scale-Consistent Learning for Partial Differential Equations Neural Operator: Graph Kernel Network for Partial Differential Equations

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.227580Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 467a4d90-31b0-4d05-b858-8e26c97e80ff · outbound

This paper cites Neural operator: Learning maps between function spaces with applications to pdes.J.

Scale-Consistent Learning for Partial Differential Equations Neural operator: Learning maps between function spaces with applications to pdes.J

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.640143Z

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 92d3083e-9bc9-4f17-8ab7-09c1ec803b1c · outbound

This paper cites Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, mar 2021.

Scale-Consistent Learning for Partial Differential Equations Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, mar 2021

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.618001Z

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 26c05ac0-7402-4da8-a0c3-856365663bd7 · outbound

This paper cites Learning operators with coupled attention.Journal of Machine Learning Research, 23(215):1–63, 2022.

Scale-Consistent Learning for Partial Differential Equations Learning operators with coupled attention.Journal of Machine Learning Research, 23(215):1–63, 2022

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.599864Z

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 41d34feb-54a8-4b88-a6a4-68cf3dcdea6a · outbound

This paper cites Convolutional neural operators.

Scale-Consistent Learning for Partial Differential Equations Convolutional neural operators

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.252925Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4ea4bcff-656d-4ce2-8f5f-ddc120604153 · outbound

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

Scale-Consistent Learning for Partial Differential Equations Fourier Neural Operator for Parametric Partial Differential Equations

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.257498Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9d4cdd8d-33f9-4822-aada-dc02dbc324da · outbound

This paper cites Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers.

Scale-Consistent Learning for Partial Differential Equations Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.261973Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 35e6179c-830b-4cda-9dd8-86a62a62c77d · outbound

This paper cites U-NO: U-shaped Neural Operators.

Scale-Consistent Learning for Partial Differential Equations U-NO: U-shaped Neural Operators

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.267721Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3a626883-14ac-4c5d-a890-ffab73f863c3 · outbound

This paper cites Physics-informed neural operator for learning partial differential equations.

Scale-Consistent Learning for Partial Differential Equations Physics-informed neural operator for learning partial differential equations

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.565155Z

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 17f5e262-8995-4230-b8e1-6e559912c0d0 · outbound

This paper cites Incorporating Symmetry into Deep Dynamics Models for Improved Generalization.

Scale-Consistent Learning for Partial Differential Equations Incorporating Symmetry into Deep Dynamics Models for Improved Generalization

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.279035Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7bcc7168-61ab-4ae3-a31f-2838b7438112 · outbound

This paper cites Lie point symmetry data augmentation for neural pde solvers.

Scale-Consistent Learning for Partial Differential Equations Lie point symmetry data augmentation for neural pde solvers

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.548987Z

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 0dfcaea3-18f9-41ad-9e0a-2ab7ab5c5756 · outbound

This paper cites Self-supervised learning with lie symmetries for partial differential equations.

Scale-Consistent Learning for Partial Differential Equations Self-supervised learning with lie symmetries for partial differential equations

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.289092Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c16a38be-42c6-4d87-bd51-7146a1919cf0 · outbound

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

Scale-Consistent Learning for Partial Differential Equations Geometry-Informed Neural Operator for Large-Scale 3D PDEs

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.294418Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 637bfc50-1ac2-4c15-8e44-e2bd1fe29e50 · outbound

This paper cites AI-aided Geometric Design of Anti-infection Catheters.

Scale-Consistent Learning for Partial Differential Equations AI-aided Geometric Design of Anti-infection Catheters

Reference 17

Resolution: verified exact; local_arxiv, observed 2026-08-15T18:15:43.010700Z

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 58ef3555-3568-4aa6-9b76-b829323d88c3 · outbound

This paper cites Scientific discovery in the age of artificial intelligence.

Scale-Consistent Learning for Partial Differential Equations Scientific discovery in the age of artificial intelligence

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.304969Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8260ba16-74a1-4033-9f38-1cfd9dbb42b4 · outbound

This paper cites Pdebench: An extensive benchmark for scientific machine learning.

Scale-Consistent Learning for Partial Differential Equations Pdebench: An extensive benchmark for scientific machine learning

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.310097Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8d6c5256-99fb-4628-8025-829da962ac2e · outbound

This paper cites Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior.Advances in Neural Information Processing Systems, 36, 2024.

Scale-Consistent Learning for Partial Differential Equations Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior.Advances in Neural Information Processing Systems, 36, 2024

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.314964Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7da480e7-b38c-44af-b6f3-d187c0ed91ba · outbound

This paper cites Multiple Physics Pretraining for Physical Surrogate Models.

Scale-Consistent Learning for Partial Differential Equations Multiple Physics Pretraining for Physical Surrogate Models

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.320682Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b2ec8de8-181d-4a51-850b-9a8f6cb14499 · outbound

This paper cites DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training.

Scale-Consistent Learning for Partial Differential Equations DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.326477Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 945a3bb7-15a2-4b13-b251-2975b17d2c20 · outbound

This paper cites UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation.

Scale-Consistent Learning for Partial Differential Equations UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.333402Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 74c78520-b942-4ae9-ae8d-c4c24e3a6aa9 · outbound

This paper cites Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs.

Scale-Consistent Learning for Partial Differential Equations Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.338532Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0329d1ce-e34c-47c5-8154-afed84037e65 · outbound

This paper cites Springer Science & Business Media, 2008.

Scale-Consistent Learning for Partial Differential Equations Springer Science & Business Media, 2008

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.467498Z

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 23cc0ac9-f3cc-4f26-b057-a62094d6b09f · outbound

This paper cites Learning homogenization for elliptic operators.SIAM Journal on Numerical Analysis, 62(4):1844–1873, 2024.

Scale-Consistent Learning for Partial Differential Equations Learning homogenization for elliptic operators.SIAM Journal on Numerical Analysis, 62(4):1844–1873, 2024

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.438932Z

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 5c3c2b7e-8c98-4e4a-8a69-c53e8b7d4d83 · outbound

This paper cites Domain decomposition methods for partial differential equations.

Scale-Consistent Learning for Partial Differential Equations Domain decomposition methods for partial differential equations

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.420243Z

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 7fb0da52-b7d6-43d1-a214-4fc4bab99d0c · outbound

This paper cites Exponential convergence for multiscale linear elliptic pdes via adaptive edge basis functions.Multiscale Modeling & Simulation, 19(2):980–1010, 2021.

Scale-Consistent Learning for Partial Differential Equations Exponential convergence for multiscale linear elliptic pdes via adaptive edge basis functions.Multiscale Modeling & Simulation, 19(2):980–1010, 2021

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.404647Z

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 c4381444-eb78-4287-b5ee-50ccb0be08c8 · outbound

This paper cites Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.

Scale-Consistent Learning for Partial Differential Equations Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.363359Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 302ecd8e-46f2-43f3-bd8f-f5bdbe651a40 · outbound

This paper cites Mosaic flows: A transferable deep learning framework for solving pdes on unseen domains.Computer Methods in Applied Mechanics and Engineering, 389:114424, 2022.

Scale-Consistent Learning for Partial Differential Equations Mosaic flows: A transferable deep learning framework for solving pdes on unseen domains.Computer Methods in Applied Mechanics and Engineering, 389:114424, 2022

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.376036Z

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 2915eb13-5f43-4b18-861d-209b8aeee297 · outbound

This paper cites Operator learning with domain decomposition for geometry generalization in pde solving.arXiv preprint arXiv:2504.00510, 2025.

Scale-Consistent Learning for Partial Differential Equations Operator learning with domain decomposition for geometry generalization in pde solving.arXiv preprint arXiv:2504.00510, 2025

Reference 31

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.373907Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 05f6e71e-4e3d-4329-9be7-ca8c97cb46aa · outbound

This paper cites Spectral Neural Operators.

Scale-Consistent Learning for Partial Differential Equations Spectral Neural Operators

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.380639Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 99b9736b-c933-4303-92cf-b991c937ad3c · outbound

This paper cites U-net: Convolutional networks for biomedical image segmentation.

Scale-Consistent Learning for Partial Differential Equations U-net: Convolutional networks for biomedical image segmentation

Reference 33

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.385850Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 60b56375-9417-44d4-a4d8-80b2a098e5c7 · outbound

This paper cites Multiwavelet-based operator learning for differential equations.

Scale-Consistent Learning for Partial Differential Equations Multiwavelet-based operator learning for differential equations

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.336632Z

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 9ca46a56-5cc5-490f-a97f-2425ec0e7711 · outbound

This paper cites Learning chaotic dynamics in dissipative systems.

Scale-Consistent Learning for Partial Differential Equations Learning chaotic dynamics in dissipative systems

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.320037Z

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 2e311954-18cb-4539-abb5-30804e511d28 · outbound

This paper cites Towards Multi-spatiotemporal-scale Generalized PDE Modeling.

Scale-Consistent Learning for Partial Differential Equations Towards Multi-spatiotemporal-scale Generalized PDE Modeling

Reference 36

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.405733Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 54bed200-3b92-4d04-a23a-fbf424f0d3c1 · outbound

This paper cites Score-based Diffusion Models in Function Space.

Scale-Consistent Learning for Partial Differential Equations Score-based Diffusion Models in Function Space

Reference 37

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.412611Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3594b8b2-fed8-4ca2-ae42-e60c97ec9bc4 · outbound

This paper cites Exponentially convergent multiscale methods for 2d high frequency heterogeneous helmholtz equations.Multiscale Modeling & Simulation, 21(3):849–883, 2023.

Scale-Consistent Learning for Partial Differential Equations Exponentially convergent multiscale methods for 2d high frequency heterogeneous helmholtz equations.Multiscale Modeling & Simulation, 21(3):849–883, 2023

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.302068Z

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 5cf2efb6-0cf3-41c6-87a6-954d065e7eb1 · outbound

This paper cites The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks.

Scale-Consistent Learning for Partial Differential Equations The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks

Reference 39

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.433864Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e6840b93-4516-45f7-afa2-66266aab54a1 · outbound

This paper cites Deep Complex Networks.

Scale-Consistent Learning for Partial Differential Equations Deep Complex Networks

Reference 40

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.446230Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d46b254c-8681-47a1-af5d-5288a3fedb37 · outbound

This paper cites SPFNO: Spectral operator learning for PDEs with Dirichlet and Neumann boundary conditions.

Scale-Consistent Learning for Partial Differential Equations SPFNO: Spectral operator learning for PDEs with Dirichlet and Neumann boundary conditions

Reference 41

Resolution: unresolved; no resolver link, observed 2026-08-15T18:15:42.452473Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 45ecc2bb-fb5a-458c-a5b0-49805795e256 · outbound

This paper cites an unresolved cited work.

Scale-Consistent Learning for Partial Differential Equations Unresolved cited work

Reference 42

Resolution: unresolved; raw_fallback, observed 2026-08-15T18:15:43.276200Z

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 f81a9aa2-1129-4502-9e4e-7427ae5aae2b · outbound

This paper cites an unresolved cited work.

Scale-Consistent Learning for Partial Differential Equations Unresolved cited work

Reference 43

Resolution: unresolved; raw_fallback, observed 2026-08-15T18:15:43.257378Z

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 ae725b28-c8be-40c2-a54a-328d46ff526a · outbound

This paper cites then we must necessarily haveΨ≡G.

Scale-Consistent Learning for Partial Differential Equations then we must necessarily haveΨ≡G

Reference 44

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T18:15:43.239156Z

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 a74640bf-48bb-45a7-9c32-75e3f9bdc6d2 · outbound

This paper cites an unresolved cited work.

Scale-Consistent Learning for Partial Differential Equations Unresolved cited work

Reference 45

Resolution: unresolved; raw_fallback, observed 2026-08-15T18:15:43.222476Z

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 5c0f4236-305d-4ad2-9a56-9a3d974ef5ea · outbound

This paper cites an unresolved cited work.

Scale-Consistent Learning for Partial Differential Equations Unresolved cited work

Reference 46

Resolution: unresolved; raw_fallback, observed 2026-08-15T18:15:43.203940Z

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 f33a18be-d8ce-44f8-91b2-f14968a185da · outbound

This paper cites The simplified version in the main text is obtained when assuming that the supervised and unsupervised contributions in (17) vanish, implying that alsoErrDM = 0, i.e.

Scale-Consistent Learning for Partial Differential Equations The simplified version in the main text is obtained when assuming that the supervised and unsupervised contributions in (17) vanish, implying that alsoErrDM = 0, i.e

Reference 47

Resolution: verified exact; raw_fallback, observed 2026-08-15T18:15:42.681138Z

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.

Pith citing papers

No inbound Pith citation observations are available.