Paper Citation Record · LEDGER

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants

As of 13 August 2026, this Paper Citation Record lists 47 of 47 outbound references and 0 inbound Pith citation observations for arXiv:2412.01232.

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

pith.paper-citation-record.v1
2412.01232 v2

Coverage vector

measured 47 of 47 reference resolution

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Pith citing papers itemized under the disclosed page cap.

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Reference resolution

47 of 47 outbound references displayed

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Outbound references

Observation d816f7b0-41b1-46c7-9135-f4cf99e9860e · outbound

This paper cites Ghoussoub, A.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Ghoussoub, A

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.408665Z

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Observation 067ae00b-34b6-4ae8-962f-4a6aa49ff22b · outbound

This paper cites Ortiz, B.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Ortiz, B

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.393540Z

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Observation 8a974b1d-8520-43c9-a1aa-701ffa4db6a2 · outbound

This paper cites Ortiz, E.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Ortiz, E

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.377596Z

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Observation 30e20a8e-cdf2-41b6-aec8-6a4e84fa4c76 · outbound

This paper cites Ortiz, L.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Ortiz, L

Reference 4

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Observation d27395b6-f95f-46bb-a258-5d12fcc53845 · outbound

This paper cites Petryk, A quasi-extremal energy principle for non-potential problems in rate-independent plasticity, Journal of the Mechanics and Physics of Solids 136 (2020) 103691.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Petryk, A quasi-extremal energy principle for non-potential problems in rate-independent plasticity, Journal of the Mechanics and Physics of Solids 136 (2020) 103691

Reference 5

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Observation 71b561a7-186a-43c6-8b52-dfe0e625874b · outbound

This paper cites Carstensen, K.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Carstensen, K

Reference 6

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Observation 111b718b-dfdc-449c-b25a-a1542b4b3114 · outbound

This paper cites Gurtin, Variational principles for linear initial-value problems, Quarterly of Applied Mathematics 22 (3) (1964) 252–256.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Gurtin, Variational principles for linear initial-value problems, Quarterly of Applied Mathematics 22 (3) (1964) 252–256

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.320174Z

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Observation 804abef9-5b34-4fee-9b1f-70bd1037c3c4 · outbound

This paper cites an unresolved cited work.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:46.304743Z

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Observation 155c8ae0-254a-4cea-bc1c-42999a6fa219 · outbound

This paper cites Acharya, Variational principle for nonlinear PDE systems via duality, Quarterly of Applied Mathematics 81 (2023) 127–140.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Acharya, Variational principle for nonlinear PDE systems via duality, Quarterly of Applied Mathematics 81 (2023) 127–140

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.290306Z

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Observation cf6e444a-fc50-4d4b-a3ec-b7de281e9590 · outbound

This paper cites Acharya, A dual variational principle for nonlinear dislocation dynamics, Journal of Elasticity 154 (1) (2023) 383–395.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Acharya, A dual variational principle for nonlinear dislocation dynamics, Journal of Elasticity 154 (1) (2023) 383–395

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.275549Z

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Observation 857f1887-4ffa-4c73-b03d-c77ee8736053 · outbound

This paper cites A Hidden Convexity in Continuum Mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants A Hidden Convexity in Continuum Mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity

Reference 11

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Observation d4b18808-0fbb-41eb-95f8-3fd31426d150 · outbound

This paper cites Singh, J.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Singh, J

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.260704Z

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Observation 3a9c28fc-f553-4412-8459-136bb5b18dff · outbound

This paper cites Acharya, B.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Acharya, B

Reference 13

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Observation a10d2bc2-91ee-4cb6-a2d7-751ef82bc282 · outbound

This paper cites Acharya, A.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Acharya, A

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.245822Z

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Observation e8831b0a-de3c-4876-9674-127e6638e445 · outbound

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:46.230740Z

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Observation 92a8bc52-3937-4906-9929-447125a8bcb6 · outbound

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:46.215316Z

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Observation 8557b2ce-b15f-4a97-9c45-6d9a3bd48881 · outbound

This paper cites Kouskiya, A.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Kouskiya, A

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.200685Z

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Observation 20707b5d-b384-4b9b-b474-74fad228c35a · outbound

This paper cites Inviscid Burgers as a degenerate elliptic problem.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Inviscid Burgers as a degenerate elliptic problem

Reference 18

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Observation 901503d8-cd0b-4755-a772-19919e526d91 · outbound

This paper cites Brenier, Hidden convexity in some nonlinear PDEs from geomety and physics, Journal of Convex Analysis 17 (3&4) (2010) 945–959.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Brenier, Hidden convexity in some nonlinear PDEs from geomety and physics, Journal of Convex Analysis 17 (3&4) (2010) 945–959

Reference 19

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 20

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:46.171902Z

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 21

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 22

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 23

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 24

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 25

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Observation 8c0c1b8c-6537-427c-9f26-0d24692197d0 · outbound

This paper cites Raissi, P.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Raissi, P

Reference 26

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 27

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Observation b83a627b-b0e2-4221-a1a4-743370c150e1 · outbound

This paper cites Cuomo, V.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Cuomo, V

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.052038Z

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Observation f2dc5dec-61c8-484f-a4df-c776202ff77b · outbound

This paper cites KAN: Kolmogorov-Arnold Networks.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants KAN: Kolmogorov-Arnold Networks

Reference 29

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This paper cites KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics

Reference 30

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Observation 90cd8d6d-373c-4b71-870a-c97f4410ae00 · outbound

This paper cites Deep Learning Alternatives of the Kolmogorov Superposition Theorem.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Deep Learning Alternatives of the Kolmogorov Superposition Theorem

Reference 31

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

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Observation 21c422ab-d8b4-49d5-a475-d8a2e0d86171 · outbound

This paper cites Sukumar, A.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Sukumar, A

Reference 33

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Observation ac232a32-e60c-4cc6-95fb-2f5cddae0da7 · outbound

This paper cites Expressivity and Approximation Properties of Deep Neural Networks with ReLU$^k$ Activation.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Expressivity and Approximation Properties of Deep Neural Networks with ReLU$^k$ Activation

Reference 34

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This paper cites Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions

Reference 35

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Observation 6953cc95-d74f-44cc-9bf1-f070a53803a3 · outbound

This paper cites de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Springer, New York, 2001.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants de Boor, A Practical Guide to Splines, Applied Mathematical Sciences, Springer, New York, 2001

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:46.015943Z

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 37

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 38

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 39

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:45.975595Z

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Observation eed585d1-ab4d-4f2c-b8cd-1fe32fdf94dc · outbound

This paper cites Sukumar, Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engi- neering 61 (12) (2004) 2159–2181.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Sukumar, Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engi- neering 61 (12) (2004) 2159–2181

Reference 40

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This paper cites Belytschko, Y.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Belytschko, Y

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:45.944666Z

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No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 42

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:45.926688Z

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Observation 2df74a33-26f8-43a1-b70c-e02f92b0a208 · outbound

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Punzo, A

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T04:40:45.912772Z

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 44

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:45.898315Z

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 45

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:45.883769Z

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Observation c5a909eb-63f8-4556-a6c9-bbfe8394d388 · outbound

This paper cites Variational Dual Solutions of Chern-Simons Theory.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Variational Dual Solutions of Chern-Simons Theory

Reference 46

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Observation 02bc82a5-6aa7-4dea-96d8-7e185d73dcbc · outbound

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Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants Unresolved cited work

Reference 47

Resolution: unresolved; raw_fallback, observed 2026-08-12T04:40:45.868384Z

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No event found in the named queried sources as of 2026-08-13T06:32:02.005865+00:00.

Pith citing papers

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