Paper Citation Record · LEDGER

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs

As of 18 August 2026, this Paper Citation Record lists 37 of 37 outbound references and 3 inbound Pith citation observations for arXiv:2508.00628.

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

pith.paper-citation-record.v1
2508.00628 v1

Coverage vector

measured 37 of 37 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-06T06:09:14.770430Z

measured 40 of 40 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-18T06:34:40.430872+00:00

measured 3 of 3 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-07-03T17:46:54.687693Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-07-03T17:48:45.310749Z

Reference resolution

37 of 37 outbound references displayed

verified exact2
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 0b1089ca-9a2a-4e4b-84f1-44738ccc4f13 · outbound

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

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

Reference 1

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Observation e8978c44-ba58-48d4-86aa-4dc6a7b08650 · outbound

This paper cites Physics- informed machine learning.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Physics- informed machine learning

Reference 2

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Observation 722f81e7-ccc1-44c9-b9cc-892c384ebf98 · outbound

This paper cites Scientific machine learning through physics-informed neural networks: Where we are and what’s next.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Scientific machine learning through physics-informed neural networks: Where we are and what’s next

Reference 3

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Observation b519d6a2-5eda-4327-84c6-d611caa18a8b · outbound

This paper cites Deepxde: A deep learning library for solving differential equations.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Deepxde: A deep learning library for solving differential equations

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-06T06:09:09.031188Z

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Observation 308de551-e662-477b-8dc6-5c288bc51ace · 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.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations

Reference 5

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Observation a3ef24d1-916d-4c6d-a6cc-a7a48c336a78 · outbound

This paper cites Physics-informed neural networks (pinns) for fluid mechanics: A review.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Physics-informed neural networks (pinns) for fluid mechanics: A review

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:22.030508Z

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Observation fea33b73-69d3-47e4-a795-68ddfd8c3bc8 · outbound

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

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

Reference 7

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Observation 15f5bad5-4d6a-4404-9c52-bd99ddfe63e6 · outbound

This paper cites Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:21.783355Z

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Observation 8007d335-39f8-42ae-b034-c5c388a1352d · outbound

This paper cites Fe-pirbn: Feature-enhanced physics- informed radial basis neural networks for solving high-frequency electromagnetic scattering problems.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Fe-pirbn: Feature-enhanced physics- informed radial basis neural networks for solving high-frequency electromagnetic scattering problems

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:21.506274Z

Source-reported events for the cited work

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Observation 3ad97f37-c64f-419a-9867-dc2a294392ae · outbound

This paper cites Physics-informed neural wavefields with gabor basis functions.Neural Networks, 177:106380, 2024.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Physics-informed neural wavefields with gabor basis functions.Neural Networks, 177:106380, 2024

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:21.218742Z

Source-reported events for the cited work

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Observation ef1dd4e3-86d7-44df-90d8-1ee89a881558 · outbound

This paper cites A physics-informed deep learning framework for inversion and surrogate modeling in solid geophysics.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs A physics-informed deep learning framework for inversion and surrogate modeling in solid geophysics

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:20.866489Z

Source-reported events for the cited work

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Observation be2d04ea-9c79-421c-b94a-d99700abbb4c · outbound

This paper cites Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:20.617694Z

Source-reported events for the cited work

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Observation 1832b70a-136f-4459-b169-6f0f7fa8a03f · outbound

This paper cites A versatile framework to solve the helmholtz equation using physics-informed neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs A versatile framework to solve the helmholtz equation using physics-informed neural networks

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:20.333011Z

Source-reported events for the cited work

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Observation be1805e0-5b6c-42de-ab81-94e16caa6118 · outbound

This paper cites Deep neural helmholtz operators for 3-d elastic wave propagation and inversion.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Deep neural helmholtz operators for 3-d elastic wave propagation and inversion

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:20.036187Z

Source-reported events for the cited work

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Observation de6c62e1-d1a8-4157-81d4-f98249c968ad · outbound

This paper cites Physics-informed neural networks for modal wave field predictions in 3d room acoustics.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Physics-informed neural networks for modal wave field predictions in 3d room acoustics

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:19.790266Z

Source-reported events for the cited work

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Observation fc251fa9-1a56-47b0-9622-1c836d7661fc · outbound

This paper cites A Neural Multigrid Solver for Helmholtz Equations with High Wavenumber and Heterogeneous Media.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs A Neural Multigrid Solver for Helmholtz Equations with High Wavenumber and Heterogeneous Media

Reference 16

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Observation eaa44a45-0ebb-4fb3-82e1-0f7e512a4540 · outbound

This paper cites Nsno: Neumann series neural operator for solving helmholtz equations in inhomogeneous medium.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Nsno: Neumann series neural operator for solving helmholtz equations in inhomogeneous medium

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:19.479914Z

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Observation 71c1d5b4-a1bc-4943-a26f-b21b847bb59c · outbound

This paper cites The Helmholtz equation with uncertainties in the wavenumber.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs The Helmholtz equation with uncertainties in the wavenumber

Reference 18

Resolution: verified exact; local_arxiv, observed 2026-08-06T06:09:15.327212Z

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Observation 9b80c536-21dd-4848-bf07-b05e15f7a99b · outbound

This paper cites Training-image based geostatistical inversion using a spatial generative adversarial neural network.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Training-image based geostatistical inversion using a spatial generative adversarial neural network

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:19.216668Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation a1614811-d025-46de-9eb3-10e984ecf95d · outbound

This paper cites On the spectral bias of neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs On the spectral bias of neural networks

Reference 20

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Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 14670fb5-6459-4835-86a4-a86a1d622162 · outbound

This paper cites On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks

Reference 21

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Observation 9a1b5a22-89b0-4d9a-aebe-27988f33f457 · outbound

This paper cites Frequency principle: Fourier analysis sheds light on deep neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Frequency principle: Fourier analysis sheds light on deep neural networks

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:18.652375Z

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No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation ffd52251-dbab-4641-bfff-f6ed911e6923 · outbound

This paper cites On understanding and overcoming spectral biases of deep neural network learning methods for solving pdes.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs On understanding and overcoming spectral biases of deep neural network learning methods for solving pdes

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:18.396265Z

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No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation cce03c48-3880-4fe0-8439-30043fc92aa0 · outbound

This paper cites Understanding and mitigating gradient flow pathologies in physics-informed neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Understanding and mitigating gradient flow pathologies in physics-informed neural networks

Reference 24

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Observation 6548d12c-4d2c-4f9b-ac58-feb6c0b1a8dd · outbound

This paper cites Characterizing possible failure modes in physics-informed neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Characterizing possible failure modes in physics-informed neural networks

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:18.138062Z

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Observation 53fd0b7d-09f8-4002-85b0-fdf1c0a2abc1 · outbound

This paper cites Fourier features let networks learn high frequency functions in low dimensional domains.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Fourier features let networks learn high frequency functions in low dimensional domains

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:17.900115Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 1bdf6119-b41a-468c-adb9-3d1384435134 · outbound

This paper cites Diminishing spectral bias in physics-informed neural networks using spatially-adaptive fourier feature encoding.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Diminishing spectral bias in physics-informed neural networks using spatially-adaptive fourier feature encoding

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:17.609794Z

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No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 1e0594cb-2217-410a-b7eb-ac89cb9fc6a6 · outbound

This paper cites Spectrum-informed multistage neural networks: Multiscale function approximators of machine precision.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Spectrum-informed multistage neural networks: Multiscale function approximators of machine precision

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:17.250144Z

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No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation cbf476e2-8fc0-4315-a5e0-860427b84f26 · outbound

This paper cites Binary structured physics-informed neural networks for solving equations with rapidly changing solutions.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Binary structured physics-informed neural networks for solving equations with rapidly changing solutions

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:16.936972Z

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No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 833d0854-2d30-4e16-96c5-9d43cebfc7f1 · outbound

This paper cites Frequency-adaptive multi-scale deep neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Frequency-adaptive multi-scale deep neural networks

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:16.606215Z

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Observation d4d761fd-436d-4337-93b4-31146c38d5c5 · outbound

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

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Fourier Neural Operator for Parametric Partial Differential Equations

Reference 31

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Source-reported events for the cited work

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Observation df6522d3-57d1-4e8e-abb2-721721c6fce6 · outbound

This paper cites Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:16.314369Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation ec7a635e-96d9-4066-b8ed-26ecd6f605b3 · outbound

This paper cites Spectral Bias in Practice: The Role of Function Frequency in Generalization.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Spectral Bias in Practice: The Role of Function Frequency in Generalization

Reference 33

Resolution: verified exact; local_arxiv, observed 2026-08-06T06:09:15.048167Z

Source-reported events for the cited work

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Observation 4d4ab669-a0c5-4ab0-9107-8e9ce7c0d17b · outbound

This paper cites On the Activation Function Dependence of the Spectral Bias of Neural Networks.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs On the Activation Function Dependence of the Spectral Bias of Neural Networks

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-06T06:09:14.324832Z

Source-reported events for the cited work

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Observation ee89359c-d15b-4db4-8f60-e9112fa2e16b · outbound

This paper cites Spectral and finite difference solutions of the burgers equation.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Spectral and finite difference solutions of the burgers equation

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:15.989922Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 75ef5a81-523d-4476-a083-f5b8e7704be5 · outbound

This paper cites Message Passing Neural PDE Solvers.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Message Passing Neural PDE Solvers

Reference 36

Resolution: unresolved; no resolver link, observed 2026-08-06T06:09:14.626065Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ea0bc1ed-751e-4c74-bc99-46853d9a3138 · outbound

This paper cites Spectral methods in MATLAB.

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs Spectral methods in MATLAB

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-06T06:09:15.701026Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Pith citing papers

Observation c6e8f305-b2a5-412c-a2e4-e1e768f177af · inbound

Gradient Scaling Effects in Adaptive Spectral PINNs for Stiff Nonlinear ODEs cites this paper.

Gradient Scaling Effects in Adaptive Spectral PINNs for Stiff Nonlinear ODEs Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs

Reference 13

Resolution: verified exact; arxiv_id, observed 2026-05-11T18:01:08.704727Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation b69062d1-dae4-4023-9497-a61d2429290e · inbound

Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs cites this paper.

Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs

Reference 49

Resolution: verified exact; arxiv_id, observed 2026-06-29T00:02:49.785835Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 48773759-fd94-40cd-91b4-497172070292 · inbound

Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations cites this paper.

Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs

Reference 31

Resolution: verified exact; arxiv_id, observed 2026-07-03T17:48:45.312294Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.