Paper Citation Record · LEDGER

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning

As of 7 August 2026, this Paper Citation Record lists 48 of 48 outbound references and 0 inbound Pith citation observations for arXiv:2607.25905.

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pith.paper-citation-record.v1
2607.25905 v1

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measured 48 of 48 reference resolution

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Source: paper_references, paper_reference_links, observed 2026-08-01T01:22:49.166462Z

measured 48 of 48 standing notices

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Source: scholarly_work_events, retraction_status_cache, observed 2026-08-06T06:34:29.942622+00:00

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Pith citing papers itemized under the disclosed page cap.

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Reference resolution

48 of 48 outbound references displayed

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Outbound references

Observation 1124f8f9-fffa-4341-b669-d8ae0f532395 · outbound

This paper cites Mathematical analysis, 1958.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Mathematical analysis, 1958

Reference 1

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Observation 09efc6b0-1a48-4eb6-ad47-7d30838fd830 · outbound

This paper cites Functions of bounded variation and free discontinuity problems.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Functions of bounded variation and free discontinuity problems

Reference 2

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Observation 2ee82292-8154-4f80-97d5-d24a302f472e · outbound

This paper cites Variational models for phase transitions, an approach via -convergence.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Variational models for phase transitions, an approach via -convergence

Reference 3

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Observation 0587230c-7d3b-42a2-bdaa-e1a8c4511475 · outbound

This paper cites Breaking the curse of dimensionality with convex neural networks.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Breaking the curse of dimensionality with convex neural networks

Reference 4

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Observation 3a57f580-4e2c-482c-a4c2-c380e4465563 · outbound

This paper cites Discontinuous equilibrium solutions and cavitation in nonlinear elasticity.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Discontinuous equilibrium solutions and cavitation in nonlinear elasticity

Reference 5

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Observation 8e59ce6d-faf1-44e0-8f88-53139ceadf1c · outbound

This paper cites Onsager's conjecture for admissible weak solutions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Onsager's conjecture for admissible weak solutions

Reference 6

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Observation 225e6be9-bfab-4a6a-8820-ff1167a93cb3 · outbound

This paper cites New examples on L avrentiev gap using fractals.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning New examples on L avrentiev gap using fractals

Reference 7

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Observation 458a7b25-9ef4-4af6-b1fc-5af5600c1866 · outbound

This paper cites Penalising the biases in norm regularisation enforces sparsity.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Penalising the biases in norm regularisation enforces sparsity

Reference 8

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Observation 56fa74bb-d603-4ace-8b83-da6d302bc141 · outbound

This paper cites A global method for relaxation in w^ 1, p and in sbv^p.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A global method for relaxation in w^ 1, p and in sbv^p

Reference 9

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Observation fa62e9b1-c9df-4fce-9cfc-818cc5bf9d46 · outbound

This paper cites Finite Element Methods for the Stretching and Bending of Thin Structures with Folding.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Finite Element Methods for the Stretching and Bending of Thin Structures with Folding

Reference 10

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Observation 6923c3e9-f127-4a3f-802b-7170876a0772 · outbound

This paper cites h-principle and rigidity for c^ 1, -isometric embeddings.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning h-principle and rigidity for c^ 1, -isometric embeddings

Reference 11

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Observation b541d1e3-502a-4d55-994a-01f3b21f3aff · outbound

This paper cites Deformation concentration for martensitic microstructures in the limit of low volume fraction.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Deformation concentration for martensitic microstructures in the limit of low volume fraction

Reference 12

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Observation 518a7e8d-3537-4a8b-bd4b-c7696425ec8e · outbound

This paper cites Onsager's conjecture on the energy conservation for solutions of E uler's equation.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Onsager's conjecture on the energy conservation for solutions of E uler's equation

Reference 13

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Observation b3373d55-39c7-4be0-bfaf-7246f2da3994 · outbound

This paper cites On turbulence and geometry: from Nash to Onsager.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On turbulence and geometry: from Nash to Onsager

Reference 14

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This paper cites Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen

Reference 15

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This paper cites Measure theory and fine properties of functions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Measure theory and fine properties of functions

Reference 16

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Observation 9ab5fd58-fd8a-4f61-b80b-d1cf16e362db · outbound

This paper cites The B arron space and the flow-induced function spaces for neural network models.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The B arron space and the flow-induced function spaces for neural network models

Reference 17

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Observation 87b5b00a-1c27-40e7-9612-e6c4b8351ded · outbound

This paper cites L’h \^o pital’s monotone rule, gromov’s theorem, and operations that preserve the monotonicity of quotients.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning L’h \^o pital’s monotone rule, gromov’s theorem, and operations that preserve the monotonicity of quotients

Reference 18

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Observation 5616e724-09e5-426b-9251-91f7b3c1f735 · outbound

This paper cites The power of depth for feedforward neural networks.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The power of depth for feedforward neural networks

Reference 19

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Observation 5e0027da-66fb-4f7e-a24d-ec49e104b187 · outbound

This paper cites Partial differential equations , volume 19.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Partial differential equations , volume 19

Reference 20

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This paper cites On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

Reference 21

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Representation formulas and pointwise properties for B arron functions

Reference 22

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Observation 6b4f7af8-29ef-41f9-bce6-708aa729dc41 · outbound

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The L avrentiev gap phenomenon in nonlinear elasticity

Reference 23

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This paper cites A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence

Reference 24

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This paper cites An anisotropic P oincar \'e inequality in gsbv^p and the limit of strongly anisotropic Mumford--Shah functionals.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning An anisotropic P oincar \'e inequality in gsbv^p and the limit of strongly anisotropic Mumford--Shah functionals

Reference 25

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Minimal surfaces and functions of bounded variation

Reference 26

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Observation 75c1494c-8609-40de-812d-31aa9072d48a · outbound

This paper cites Elliptic partial differential equations of second order , volume 224.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Elliptic partial differential equations of second order , volume 224

Reference 27

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Sur quelques problemes du calcul des variations

Reference 28

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On the L avrentiev phenomenon

Reference 29

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Strong approximation of special functions of bounded variation functions with prescribed jump direction

Reference 30

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory

Reference 31

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Unresolved cited work

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The L avrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

Reference 33

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning C ^1 -isometric imbeddings

Reference 34

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The imbedding problem for R iemannian manifolds

Reference 35

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This paper cites A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

Reference 36

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Banach space representer theorems for neural networks and ridge splines

Reference 37

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Minimum norm interpolation by perceptra: Explicit regularization and implicit bias

Reference 38

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Real and complex analysis

Reference 39

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Benefits of depth in neural networks

Reference 40

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Ridges, neural networks, and the radon transform

Reference 41

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Depth separation beyond radial functions

Reference 42

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective

Reference 43

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Lipschitz algebras

Reference 44

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality

Reference 45

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A note on elliptic regularity theory in barron spaces

Reference 46

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This paper cites Averaging of functionals of the calculus of variations and elasticity theory.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Averaging of functionals of the calculus of variations and elasticity theory

Reference 47

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The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Lavrentiev phenomenon and homogenization for some variational problems

Reference 48

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