Paper Citation Record · LEDGER

A Variational Framework for the Complexity of PDE Solutions

As of 18 August 2026, this Paper Citation Record lists 49 of 49 outbound references and 1 inbound Pith citation observation for arXiv:2510.21290.

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

pith.paper-citation-record.v1
2510.21290 v3

Coverage vector

measured 49 of 49 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-04T08:24:46.338304Z

measured 50 of 50 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 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-04T09:40:32.886359Z

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

49 of 49 outbound references displayed

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Outbound references

Observation 7cb581b1-2b9e-42a9-9400-2d9f6c987b61 · outbound

This paper cites On computable numbers, with an application to the entscheidungsproblem.J.

A Variational Framework for the Complexity of PDE Solutions On computable numbers, with an application to the entscheidungsproblem.J

Reference 1

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Observation c48c8a78-690e-4df7-99a6-677eddecadfe · outbound

This paper cites Monte carlo solution of partial differential equations using a hybrid computer.IEEE Trans.

A Variational Framework for the Complexity of PDE Solutions Monte carlo solution of partial differential equations using a hybrid computer.IEEE Trans

Reference 2

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Observation ae55c05f-e50e-40f9-ac6a-4c36807ba38b · outbound

This paper cites The wave equation with computable ini- tial data whose unique solution is nowhere computable.MLQ Math.

A Variational Framework for the Complexity of PDE Solutions The wave equation with computable ini- tial data whose unique solution is nowhere computable.MLQ Math

Reference 3

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Observation 72bfc2ea-77fa-4325-9b1f-a7cafa4471c6 · outbound

This paper cites Computing schr¨ odinger propagators on type-2 turing machines.J.

A Variational Framework for the Complexity of PDE Solutions Computing schr¨ odinger propagators on type-2 turing machines.J

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:42.580652Z

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Observation 90576eee-2616-4fc9-a630-34318159796a · outbound

This paper cites Complexity blowup for solutions of the laplace and the diffusion equation, 2023.

A Variational Framework for the Complexity of PDE Solutions Complexity blowup for solutions of the laplace and the diffusion equation, 2023

Reference 5

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Observation b3040939-0a00-432c-86b8-99d65bfdc4aa · outbound

This paper cites American Mathematical Society, Providence, RI, March 2022.

A Variational Framework for the Complexity of PDE Solutions American Mathematical Society, Providence, RI, March 2022

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:42.770970Z

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Observation 86699286-73c7-43ff-8115-22cfbce7c7f9 · outbound

This paper cites Soft- ware, environments, tools.

A Variational Framework for the Complexity of PDE Solutions Soft- ware, environments, tools

Reference 7

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Observation cdcabe87-f751-4f42-b3c4-da6cc683c6d9 · outbound

This paper cites Least- squares space-time formulation for advection-diffusion problem with efficient adaptive solver based on matrix compression.

A Variational Framework for the Complexity of PDE Solutions Least- squares space-time formulation for advection-diffusion problem with efficient adaptive solver based on matrix compression

Reference 8

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Observation 01230a5e-03f9-4e40-9dd2-28f7b385867c · outbound

This paper cites Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs.J.

A Variational Framework for the Complexity of PDE Solutions Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs.J

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:43.040096Z

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Observation 07f7587c-b52c-41ec-a046-c09bd0d8979e · outbound

This paper cites Tackling the curse of dimensionality with physics-informed neural networks.

A Variational Framework for the Complexity of PDE Solutions Tackling the curse of dimensionality with physics-informed neural networks

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:43.132996Z

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Observation 22bd4767-de64-4781-9402-962219c08172 · outbound

This paper cites Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power.Mach.

A Variational Framework for the Complexity of PDE Solutions Polynomial differentiation decreases the training time complexity of physics-informed neural networks and strengthens their approximation power.Mach

Reference 11

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Observation f32dcc43-5ee3-4d23-b5d9-d3db5c9bc712 · outbound

This paper cites Dissertation, Technische Universit¨ at Dresden, June 2025.

A Variational Framework for the Complexity of PDE Solutions Dissertation, Technische Universit¨ at Dresden, June 2025

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:43.290173Z

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Observation 3195eee4-64b2-4c4d-ba49-6f14ef287a3e · outbound

This paper cites A variational principle for gradient flows.Math.

A Variational Framework for the Complexity of PDE Solutions A variational principle for gradient flows.Math

Reference 13

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Observation 256e5d6f-57ab-4c24-a331-b8662b76c459 · outbound

This paper cites Lectures in Mathematics.

A Variational Framework for the Complexity of PDE Solutions Lectures in Mathematics

Reference 14

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Observation 9a7679cf-a07d-4b31-88ea-aa52b5102a03 · outbound

This paper cites Cambridge University Press, Cambridge, England, June 2012.

A Variational Framework for the Complexity of PDE Solutions Cambridge University Press, Cambridge, England, June 2012

Reference 16

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Observation 8c15170a-4104-41fa-a6fd-82ae9473f269 · outbound

This paper cites Noncomputability in analysis and physics: A complete determination of the class of noncomputable linear operators.

A Variational Framework for the Complexity of PDE Solutions Noncomputability in analysis and physics: A complete determination of the class of noncomputable linear operators

Reference 17

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Observation 1d45c0ff-2406-47a8-8ca3-615c0fd64293 · outbound

This paper cites Turing meets circuit theory: Not every continuous- time LTI system can be simulated on a digital computer.IEEE Trans.

A Variational Framework for the Complexity of PDE Solutions Turing meets circuit theory: Not every continuous- time LTI system can be simulated on a digital computer.IEEE Trans

Reference 18

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Observation b406ddb0-ed29-4b16-87d4-0e85519ab251 · outbound

This paper cites Is the linear schr¨ odinger propagator turing computable? InComputability and Complexity in Analysis, Lecture notes in computer science, pages 369–377.

A Variational Framework for the Complexity of PDE Solutions Is the linear schr¨ odinger propagator turing computable? InComputability and Complexity in Analysis, Lecture notes in computer science, pages 369–377

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:43.957760Z

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Observation 26d836d0-98cd-4d3f-bbe2-23aed5efb3f1 · outbound

This paper cites On the computational complexity of the dirichlet problem for poisson’s equation.Math.

A Variational Framework for the Complexity of PDE Solutions On the computational complexity of the dirichlet problem for poisson’s equation.Math

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:44.022732Z

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Observation cf8f4967-de12-4e61-8a12-ec6f6c18fac1 · outbound

This paper cites Bit- complexity of solving systems of linear evolutionary partial differential equations.

A Variational Framework for the Complexity of PDE Solutions Bit- complexity of solving systems of linear evolutionary partial differential equations

Reference 21

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Observation 2155d9b6-0ae5-4789-aad8-344dc9c041b6 · outbound

This paper cites Cambridge University Press, Cambridge, England, April 2009.

A Variational Framework for the Complexity of PDE Solutions Cambridge University Press, Cambridge, England, April 2009

Reference 22

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Observation 435200a1-1c43-4272-a55d-377672e267a4 · outbound

This paper cites Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness.Mach.

A Variational Framework for the Complexity of PDE Solutions Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness.Mach

Reference 23

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Observation 6716d341-b06c-4387-9c0b-4b5cfe5d0352 · outbound

This paper cites Linear convergence of gra- dient and proximal-gradient methods under the Polyak- Lojasiewicz condition.

A Variational Framework for the Complexity of PDE Solutions Linear convergence of gra- dient and proximal-gradient methods under the Polyak- Lojasiewicz condition

Reference 24

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Observation d6c8f735-fc84-4043-9b51-0b66172ccffb · outbound

This paper cites Linear convergence of gradient and proximal-gradient methods under the polyak- lojasiewicz condition, 2020.

A Variational Framework for the Complexity of PDE Solutions Linear convergence of gradient and proximal-gradient methods under the polyak- lojasiewicz condition, 2020

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-04T08:24:44.442074Z

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Observation 51f4dae7-2839-4528-bea1-eabc4ad38fe5 · outbound

This paper cites Trefethen.Approximation theory and approximation practice, volume.

A Variational Framework for the Complexity of PDE Solutions Trefethen.Approximation theory and approximation practice, volume

Reference 26

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Observation 9e9e7aa7-e694-4b9f-b29e-87a5739c36c6 · outbound

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A Variational Framework for the Complexity of PDE Solutions Unresolved cited work

Reference 27

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Observation 6f984c91-1a56-4dfe-9c80-3555b6e2d4dc · outbound

This paper cites Multivariate polynomial approximation in the hypercube.

A Variational Framework for the Complexity of PDE Solutions Multivariate polynomial approximation in the hypercube

Reference 28

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Observation dc335d05-6902-4417-a081-c44aeac4792f · outbound

This paper cites High-order integration on regular triangulated manifolds reaches super-algebraic approximation rates through cubical re-parameterizations, 2024.

A Variational Framework for the Complexity of PDE Solutions High-order integration on regular triangulated manifolds reaches super-algebraic approximation rates through cubical re-parameterizations, 2024

Reference 29

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Observation 3b1b129b-c804-4894-b877-db0f42edec3a · outbound

This paper cites Calculation of gauss quadrature rules.Math.

A Variational Framework for the Complexity of PDE Solutions Calculation of gauss quadrature rules.Math

Reference 30

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Observation 0f3ac7db-8599-4277-b6b9-45e63ac3060e · outbound

This paper cites Perspectives in Mathematical Logic.

A Variational Framework for the Complexity of PDE Solutions Perspectives in Mathematical Logic

Reference 31

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Observation c97b72b6-859f-4ead-855e-082038db74d7 · outbound

This paper cites On the definitions of computable real continuous functions.

A Variational Framework for the Complexity of PDE Solutions On the definitions of computable real continuous functions

Reference 32

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Observation fad674c8-73ed-453b-ade6-e83ca998065a · outbound

This paper cites McGraw-Hill series in higher mathematics.

A Variational Framework for the Complexity of PDE Solutions McGraw-Hill series in higher mathematics

Reference 33

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Observation 957cf23f-7dad-4e97-aaa7-77a50b169be8 · outbound

This paper cites The direct method in the calculus of variations.

A Variational Framework for the Complexity of PDE Solutions The direct method in the calculus of variations

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Observation 5ef9f62d-6225-49a5-b5d9-53401b1d90af · outbound

This paper cites Springer, 2011.

A Variational Framework for the Complexity of PDE Solutions Springer, 2011

Reference 35

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Observation 1903106f-0c3c-4c30-98f6-aecfba6e1d17 · outbound

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A Variational Framework for the Complexity of PDE Solutions Applied mathematical sciences

Reference 36

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Observation 737244c8-5e79-4a04-b16e-311b6b24dccb · outbound

This paper cites Compatibility conditions for dirichlet and neumann problems of poisson’s equation on a rectangle.J.

A Variational Framework for the Complexity of PDE Solutions Compatibility conditions for dirichlet and neumann problems of poisson’s equation on a rectangle.J

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Observation 697b72db-9178-4466-bb29-6e0081375d88 · outbound

This paper cites Analysis of fractional differential equations.J.

A Variational Framework for the Complexity of PDE Solutions Analysis of fractional differential equations.J

Reference 38

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Observation c365c0e5-8483-457f-af9d-ad347cdb7e1a · outbound

This paper cites Hypoelliptic operators.

A Variational Framework for the Complexity of PDE Solutions Hypoelliptic operators

Reference 39

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Observation 8967d73e-f437-41b8-913e-f77b0a0fb48a · outbound

This paper cites Singularities of solutions of the eikonal equation.Differ.

A Variational Framework for the Complexity of PDE Solutions Singularities of solutions of the eikonal equation.Differ

Reference 40

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Observation 87f92392-cfa3-4597-9394-3469d05242f5 · outbound

This paper cites Revisiting subgradient method: Complexity and convergence beyond lipschitz continuity.

A Variational Framework for the Complexity of PDE Solutions Revisiting subgradient method: Complexity and convergence beyond lipschitz continuity

Reference 41

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Observation bcdaf464-76ed-48ff-b707-cc9c4e5f587a · outbound

This paper cites Springer International Publishing, 2015 edition, January 2014.

A Variational Framework for the Complexity of PDE Solutions Springer International Publishing, 2015 edition, January 2014

Reference 42

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Observation a364f81a-88be-44d0-8745-029eefc510ae · outbound

This paper cites Golub and John H.

A Variational Framework for the Complexity of PDE Solutions Golub and John H

Reference 43

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Observation e9033be1-ae08-41fd-b940-38362600eade · outbound

This paper cites Fast and accurate computation of Gauss– Legendre and Gauss–Jacobi quadrature nodes and weights.SIAM J.

A Variational Framework for the Complexity of PDE Solutions Fast and accurate computation of Gauss– Legendre and Gauss–Jacobi quadrature nodes and weights.SIAM J

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Observation 99a1cead-1a75-4859-9a5e-fe5f7dd1b935 · outbound

This paper cites On the computation of the Gauss–Legendre quadrature formula with a given precision.J.

A Variational Framework for the Complexity of PDE Solutions On the computation of the Gauss–Legendre quadrature formula with a given precision.J

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Observation 6e886cc2-54a8-459d-b08b-a13b14fb959b · outbound

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A Variational Framework for the Complexity of PDE Solutions Unresolved cited work

Reference 46

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Observation 57abfa14-0eb8-4d16-98b7-41ed6839b564 · outbound

This paper cites More special functions.

A Variational Framework for the Complexity of PDE Solutions More special functions

Reference 47

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Observation bd0929b9-0cf7-4f39-a514-14eda05d916d · outbound

This paper cites Springer monographs in mathematics.

A Variational Framework for the Complexity of PDE Solutions Springer monographs in mathematics

Reference 48

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Observation c6425992-120d-4ed0-b987-36b96b4cafc9 · outbound

This paper cites Traces of sobolev functions on fractal type sets and characterization of extension domains.Journal of Functional Analysis, 143(1):221–246, 1997.

A Variational Framework for the Complexity of PDE Solutions Traces of sobolev functions on fractal type sets and characterization of extension domains.Journal of Functional Analysis, 143(1):221–246, 1997

Reference 49

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Observation 12d5bb62-a5c9-4e87-b97c-441b4f2484c6 · outbound

This paper cites Birkh¨ auser Boston, Boston, MA, 1993.

A Variational Framework for the Complexity of PDE Solutions Birkh¨ auser Boston, Boston, MA, 1993

Reference 50

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Pith citing papers

Observation b34e6494-1782-458d-b8f4-d73f1dbdad7b · inbound

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes cites this paper.

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes A Variational Framework for the Complexity of PDE Solutions

Reference 17

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