Paper Citation Record · LEDGER

Numerical PDE solvers outperform neural PDE solvers

As of 15 August 2026, this Paper Citation Record lists 14 of 14 outbound references and 1 inbound Pith citation observation for arXiv:2507.21269.

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

pith.paper-citation-record.v1
2507.21269 v1

Coverage vector

measured 14 of 14 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T13:03:54.398768Z

measured 15 of 15 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-15T06:32:42.880941+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-07-03T02:29:11.422793Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-07-03T02:37:34.129389Z

Reference resolution

14 of 14 outbound references displayed

verified exact0
verified fuzzy7
unresolved6
parse uncertain0
malformed identifier0
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 5eb5d9e0-9b39-434c-b5de-5997eb3e5160 · outbound

This paper cites On the convergence and generalization of physics informed neural networks.

Numerical PDE solvers outperform neural PDE solvers On the convergence and generalization of physics informed neural networks

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:55.894765Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 8e16ecc6-f82f-4a35-8c05-aab095ec9c0f · outbound

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

Numerical PDE solvers outperform neural PDE solvers Pdebench: An extensive benchmark for scientific machine learning

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:55.688395Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 445a895a-06ba-4528-823c-6b259d86a5f7 · outbound

This paper cites Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers.

Numerical PDE solvers outperform neural PDE solvers Solver-in-the- loop: Learning from differentiable physics to interact with iterative pde-solvers

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:55.489930Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation a9925d72-dea3-49c1-bf28-2ead38f136e7 · outbound

This paper cites Climode: Climate and weather forecasting with physics-informed neural odes.

Numerical PDE solvers outperform neural PDE solvers Climode: Climate and weather forecasting with physics-informed neural odes

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:55.286101Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation f70747f5-5643-4f3e-acf7-8136442abf23 · outbound

This paper cites Zhang, X.

Numerical PDE solvers outperform neural PDE solvers Zhang, X

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:54.398768Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f1805444-5c22-49a1-be03-3328ee835bb4 · outbound

This paper cites Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators.

Numerical PDE solvers outperform neural PDE solvers Solving forward and inverse PDE problems on unknown manifolds via physics-informed neural operators

Reference 2006

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:52.744566Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 34500562-00d3-450f-8d7d-346a65d203dd · outbound

This paper cites Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics.

Numerical PDE solvers outperform neural PDE solvers Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

Reference 2012

Resolution: metadata mismatch; local_arxiv, observed 2026-08-06T13:03:54.769597Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 3b133e11-96ba-497d-8b0b-0a2bd15c3e3a · outbound

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

Numerical PDE solvers outperform neural PDE solvers U-net: Convolutional networks for biomedical image segmentation

Reference 2017

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:56.090832Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation e2df9d41-ba6b-48dd-ab07-3180bfb07b32 · outbound

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

Numerical PDE solvers outperform neural PDE solvers Fourier Neural Operator for Parametric Partial Differential Equations

Reference 2019

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:53.017686Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ef8abbbd-263b-498f-81da-d665cb934314 · outbound

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

Numerical PDE solvers outperform neural PDE solvers Neural Operator: Graph Kernel Network for Partial Differential Equations

Reference 2020

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:53.185377Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 46769c8a-7393-45ba-a7ee-7dbb00ef0641 · outbound

This paper cites DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators.

Numerical PDE solvers outperform neural PDE solvers DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

Reference 2021

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:52.865638Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4d2838fc-e670-4790-b5fb-f436ff28b167 · outbound

This paper cites Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit.

Numerical PDE solvers outperform neural PDE solvers Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit

Reference 2022

Resolution: unresolved; no resolver link, observed 2026-08-06T13:03:53.945985Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8b0399bb-f6e7-4125-a5d6-f61d58a00103 · outbound

This paper cites [CORJO23] L.

Numerical PDE solvers outperform neural PDE solvers [CORJO23] L

Reference 2023

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:56.320907Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation cb3248a2-e6dd-4124-b48e-b10c9d956b0c · outbound

This paper cites [ZL W22] Qingqing Zhao, David B.

Numerical PDE solvers outperform neural PDE solvers [ZL W22] Qingqing Zhao, David B

Reference 2024

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T13:03:55.125199Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Pith citing papers

Observation b4f37434-43b9-4dda-ae45-1306ff07be89 · inbound

Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots cites this paper.

Identifiability Limits of Physics-Informed Inference for Spatial Stochastic Dynamics from Static Snapshots Numerical PDE solvers outperform neural PDE solvers

Reference 43

Resolution: verified exact; arxiv_id, observed 2026-07-03T02:37:34.130972Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.