A Unified Variational Framework for Optimal Transport with Lagrangian Costs

Hailiang Liu

REVIEW 4 major objections 5 minor 37 references

A Unified Variational Framework for Optimal Transport with Lagrangian Costs

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For any Tonelli Lagrangian, five formulations of a transport distance coincide.

desk verdict Organized repackaging of classical equivalences; true but not new, and the main proof has an unverified duality gap. read the letter →

arxiv 2607.15471 v1 pith:C6NKC334 submitted 2026-07-16 math.OC

Hailiang Liu This is my paper · ORCID

classification math.OC MSC 49K2049L2049Q20

keywords optimaltransportLagrangiancostTonelliconvexdualityHamilton-JacobiequationsWasserstein-HamiltonianflowdistanceEulerianformulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that optimal transport with a general Lagrangian cost—not just the quadratic one—has a fully unified variational description. For smooth densities on a flat torus and any Tonelli Lagrangian, five (and with strong convexity, six) apparently distinct formulations of the induced distance coincide: trajectory, Eulerian, convex momentum, Hamilton–Jacobi dual, and Hamiltonian flow. The reason they coincide is a pointwise Legendre-duality lemma that converts the action integrand into a linear functional over a convex set, plus a zero-duality-gap argument. A sympathetic reader would care because it extends the classic dynamical reformulation of the quadratic Wasserstein distance to arbitrary mechanical and optimal-control actions, and it makes the optimality system a coupled Hamilton–Jacobi/continuity PDE pair with a natural Poisson structure on the space of densities. The key load-bearing step is the claimed zero duality gap in the sup–inf interchange; the paper asserts it follows from standard convex duality without checking the relevant constraint qualification in detail.

What carries the argument

The engine is Lemma 3.1: for each point x, define the Hamiltonian H(x,p)=sup_q(p·q − L(x,q)) and the convex set Q(x)={(a,b) : a+H(x,b)≤0}. Then for any (τ,y), sup_{(a,b)∈Q(x)} (aτ+b·y)=τ L(x,y/τ) (with the convention 0 for (0,0) and +∞ elsewhere). This identity rewrites the action density ρ L(x,m/ρ) as a supremum over linear functions, turning the nonconvex Eulerian problem into a convex one and leading via Legendre duality to the Hamilton–Jacobi constraint. The zero-duality-gap step in Theorem 3.2 and the elliptic projection in Theorem 3.3 are secondary mechanisms that close the equivalence chain.

What would settle it

Take the circle T, L(x,v)=|v|^4/4, and smooth normalized densities ρ0=1+0.1 cos(2πx), ρ1=1−0.1 cos(2πx). The paper predicts I4=I5=IL; compute IL via direct discretization of the plan problem and I5 via the supremum over φ with ∂tφ + (3/4)|∂xφ|^{4/3} ≤ 0. Agreement to numerical tolerance confirms the duality, while any gap beyond solver error would falsify the zero-duality-gap assertion.

Watch

Extended reading notes

Core claim

The paper proves that when the cost is the least action of a Tonelli Lagrangian—one that is strictly convex and grows superlinearly in velocity, with complete Euler–Lagrange flow—the induced transport distance between two smooth densities on the torus can be computed in any of five (six with strong convexity) equivalent ways: (1) minimizing over particle trajectories, (2) Eulerian density–velocity fields, (3) a convex problem in density–momentum, (4) a supremum over potentials satisfying a Hamilton–Jacobi inequality, and (5) an infimum over density–phase pairs obeying a Hamiltonian flow. The proof goes through four steps: Eulerian configurations dominate trajectory configurations; the Euleri

Load-bearing premise

The central load-bearing premise is that the convex momentum infimum and the Hamilton–Jacobi supremum have zero duality gap, asserted from a general convex-duality theorem without verifying the interior feasibility condition; if a gap exists for some admissible density–momentum pair, the equality I4=I5, and hence the full chain, collapses.

Editorial extensions

If this is right

For computation, any of the equivalent formulations gives the same number, so a practitioner can choose the most tractable form—typically the convex momentum problem or the Hamilton–Jacobi supremum.
The optimality conditions for the transport problem are exactly the coupled Hamilton–Jacobi and continuity equations, ∂tφ + H(x,∇φ)=0 and ∂tρ + ∇·(ρ∇pH(x,∇φ))=0, with density endpoints prescribed; this gives a PDE characterization of optimal transport.
The induced distance inherits a formal Hamiltonian structure on the space of probability densities, with Hamiltonian ∫ H(x,∇φ)ρ dx; dynamics on the density manifold follow Hamilton's equations.
The framework covers quadratic, kinetic-plus-potential, anisotropic quadratic, and optimal-control-induced Lagrangians, giving each a canonical transport metric and connecting to mean-field games and continuous-depth network training.
The equivalence implies that the formal Riemannian geodesic distance I1 agrees with all Eulerian formulations for smooth densities, so the metric geometry of the density manifold is consistent across perspectives.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: If the zero-duality-gap assertion truly holds, a natural testable extension is to non-smooth densities and non-Tonelli but convex Lagrangians; failure there would localize the breakpoint of the chain without affecting the Eulerian–Lagrangian equality.
Editorial: The elliptic projection in Theorem 3.3 suggests a general recipe—project any admissible momentum onto a velocity-gradient form—which could yield a canonical gauge choice for velocity fields in numerical Eulerian schemes for non-quadratic costs.
Editorial: The Hamiltonian-flow formulation invites a concrete numerical experiment comparing I4 and I6 values for a non-quadratic strongly convex Lagrangian on T², which would either confirm the elliptic projection step or expose the regularity gap behind it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the punchline: the paper is a clean, well-cited catalog of known equivalences among Lagrangian, Eulerian, convex, and Hamilton–Jacobi formulations of Lagrangian optimal transport on the torus. What is actually new is modest: a numbering scheme that puts six forms side by side and a direct Hamiltonian-flow formulation I6 obtained by substituting the first-order condition into the convex problem. The author knows the literature—Bernard–Buffoni, Fathi–Figalli, Villani, Ambrosio–Gigli–Savaré are all there—and the examples are correct.

The soft spots are where the proof leans on assertions. In Theorem 3.2, Step 3, the zero-duality-gap is announced and then a smooth solution to the coupled HJ/continuity system is assumed; the Fenchel–Rockafellar constraint qualification is never checked. That is load-bearing for I4=I5, and the alternative inequality route in Step 4 is sketched, not completed. Theorem 3.3's elliptic step is similarly a one-line 'standard elliptic regularity,' which may be fine for smooth data but needs a statement. The geometric formulation I1 is introduced but never proved equivalent; the Discussion overclaims it. There are also numerous typos in the displays that obscure the steps.

These are real gaps, but they are the kind a referee could force the author to either fix or label as formal. The central equivalences are true in the literature, so the paper is not building on sand; it is just not a new theorem. The value is organizational, not foundational.

Who is this for? A reader who wants one place to see how the Lagrangian, Eulerian, convex, and HJ viewpoints align, and who is willing to track down the original proofs. For a research contribution, it needs a clear statement of what is new relative to [6], [19], and [37], and a complete duality argument. I would not desk reject it—a referee can help sort that out—but I would send it back for major revision, not accept the current version.

Recommendation: engage with it if the author is willing to be honest about the overlap; otherwise, it is a survey note.

Referee Report

4 major / 5 minor

Summary. The paper studies optimal transport on the flat torus for costs induced by a Tonelli Lagrangian L(x,v) via the action c(x,y)=inf∫L(ω,ω̇)dt. It proposes six variational formulations: a formal Riemannian/geometric distance I1, a Lagrangian flow formulation I2, an Eulerian formulation I3, a convex momentum formulation I4 obtained by the Benamou–Brenier change of variables m=ρv, a Hamilton–Jacobi dual formulation I5, and a Hamiltonian flow formulation I6. Theorem 3.2 claims IL=I2=I3 and IL=I4=I5; Theorem 3.3 claims IL=I6 under strong convexity. The proof strategy is to prove I3≥IL, convexify to I4, use Fenchel–Rockafellar duality to identify I4 with I5, and then construct admissible (ρ,m) from an optimal plan to close the chain. Examples include quadratic cost, mechanical Lagrangians, translation-invariant costs, anisotropic quadratic costs, and control/neural-ODE settings.

Significance. If fully established, the paper would provide a unified presentation of several known equivalences for Lagrangian optimal transport and would connect them to Wasserstein–Hamiltonian flows and mean-field-game-type systems. The individual statements are consistent with results in the literature (e.g., Bernard–Buffoni, Fathi–Figalli), so the conceptual contribution is mainly organizational, with the I6 Hamiltonian reformulation being the clearest potentially new element. The paper contains no computational artifacts; its value would be a clean synthetic framework. However, the proof as written has substantial gaps in exactly the steps needed to make the framework rigorous, so the contribution is currently conditional.

major comments (4)

[Theorem 3.2, Step 1 (pp. 8–9)] The proof does not establish the claimed equalities IL=I2=I3. It shows that every admissible Eulerian pair has cost at least ∫ρ0 c(x,X(1,x))dx ≥ IL, i.e., I3≥IL, and similarly I2≥IL. No reverse inequality is provided. To prove IL≤I3 one needs a representation/approximation of an optimal transport plan by smooth Eulerian flows (e.g., superposition principle or the generalized Benamou–Brenier theorem). This is a load-bearing omission for the theorem's first assertion.
[Theorem 3.2, Step 3, Eq. (3.11)] The zero-duality-gap assertion is not proved. The text invokes Fenchel–Rockafellar duality but does not verify constraint qualification for the functional J in (3.9) on a suitable Banach space. The attempted verification instead assumes existence of a global smooth solution to the coupled system with ∂tφ+H(x,∇φ)=0 and only ρ endpoint data. For a Tonelli Hamiltonian on T^d such classical HJ solutions generally do not exist globally for arbitrary smooth ρ0,ρ1; the canonical Kantorovich potentials for quadratic cost are often only semiconcave, not C^1. Thus the equality I4=I5 is unsupported as written.
[Theorem 3.2, Step 4 (pp. 11–13)] The argument intended to prove IL≤supφ K(φ) is logically reversed. The derivation shows that every C^1 HJ subsolution yields an admissible Kantorovich pair, which implies the supremum over HJ subsolutions is ≤ the Kantorovich supremum, i.e., sup K ≤ IL. To obtain IL≤sup K one must prove that every optimal Kantorovich pair can be represented by an (even nonsmooth) HJ subsolution; this is not supplied. The superposition construction also produces measure-valued ρ(t,x), while J in (3.9) is initially defined for densities; the passage to measures requires justification or approximation.
[Theorem 3.3, Eq. (3.13)] The proof of IL=I6 relies on solving, for each t, the nonlinear elliptic problem ∇·(m−ρ∇pH(x,∇φ))=0 on the torus and asserting existence and smoothness via 'standard elliptic regularity'. Existence of a periodic solution, the choice of additive constant, and smooth dependence on t are not demonstrated. These are nontrivial for a general Tonelli Lagrangian; without them the converse inequality I6≤I4 is not established.

minor comments (5)

[§3, Theorem 3.2 statement] The text says 'We now show that all of the preceding formulations are equivalent,' but the theorem omits I1 (the geometric/Riemannian formulation). Either include I1 in the theorem or rephrase the introduction and discussion.
[Step 3, boundary term] In the final displayed equation of Step 3, the boundary term is written as ∫(φ(1)ρ1 − ρ(0)ρ0)dx; the second term should be φ(0)ρ0.
[Step 4, notation] The function dH is defined inconsistently: earlier as L+H−v·ξ, later as L−v·ξ+H−v·ξ. This typo makes the Legendre identity in Step 4 hard to follow.
[Throughout] The regularity class of φ in the dual formulation K(φ) is not specified (C^1 vs. viscosity solution). Also, the domain of J in (3.9) should be stated precisely to distinguish densities from measures.
[Typos] Several typographical errors: 'Lebesque' should be 'Lebesgue'; 'preformed' should be 'performed'; in reference [13] 'Di!erential' should be 'Differential'; in the definition of I6, ∂tρ+∇x(ρ∇pH) should be ∂tρ+∇x·(ρ∇pH).

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the variational equivalences are genuine two-sided identities, with only a minor non-load-bearing self-citation.

full rationale

The central derivation is not circular. The quantities IL, I2, I3, I4, I5, and I6 are independently defined variational problems, and Theorems 3.2-3.3 prove equality by establishing inequalities in both directions rather than defining one quantity as another. Lemma 3.1 is a direct Legendre-transform computation. Step 3's appeal to Fenchel-Rockafellar duality is an invocation of an external theorem; the unverified constraint qualification / smooth-solution existence is a rigor gap, not a circular reduction. Step 4 constructs admissible (rho,m) from an optimal plan and uses the dual representation to prove IL >= inf J, then proves the reverse inequality through Kantorovich duality; this is a genuine derivation. The Hamiltonian formulation I6 is introduced via the substitution m = rho * grad_p H(x,grad phi), but equality with I4 is then proved from the elliptic problem (3.13), the orthogonality identity, and Fenchel-Young, so it is not merely a renaming or an ansatz. The only self-citation is reference [28], used in Example 4 as an example of anisotropic Fokker-Planck gradient flows; it is not load-bearing for any theorem. Accordingly, no specific circular step can be exhibited, and the score reflects only the presence of a minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standing Tonelli framework and smoothness hypotheses plus several unproved structural assumptions (zero duality gap, solvability of the HJ/continuity and elliptic problems). None of these are fitted to data; they are the price of a formal proof. The paper's contribution is therefore bounded by these assumptions, all of which are standard in the cited literature except the asserted elliptic solvability in Theorem 3.3.

assumptions (6)

domain assumption L is a Tonelli Lagrangian: L(x,·) strictly convex and superlinear, Euler-Lagrange flow complete
§2 defines cost c(x,y)=inf∫L and guarantees existence of minimizing curves. Standard but a standing assumption of the whole paper.
domain assumption ρ0, ρ1 are smooth probability densities (strictly positive in Theorem 3.3)
Theorem 3.2/3.3 hypotheses; smoothness is used for flows, change of variables, and elliptic regularity.
domain assumption An optimal transport plan π for the Kantorovich problem exists and minimizers X(t;x,y) can be selected measurably
Step 4 of Theorem 3.2 constructs the superposition (ρ,m) from π and minimizers; asserted ('One readily verify').
ad hoc to paper Zero duality gap for the convex momentum program (Fenchel-Rockafellar constraint qualification)
Step 3: 'the duality relation (3.11) follows directly from the Fenchel–Rockafellar duality theorem' — asserted, not verified; not strictly needed if Step 4's chain is completed.
ad hoc to paper Smooth solutions exist for the coupled HJ/continuity system, and for the elliptic problem (3.13) on T^d
Step 3 and Theorem 3.3 rely on 'standard elliptic regularity'; the elliptic problem is nonlinear and its solvability needs ρ ≥ c > 0 and coercivity, never stated as hypotheses.
domain assumption Strong convexity of L(x,·) for Theorem 3.3
Used to pass from convexity to the ellipticity of (3.13).

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Unified Variational Framework for Optimal Transport with Lagrangian Costs." pith.science (2026). https://pith.science/paper/C6NKC334

@misc{pith2026260715471,
  author       = {Pith},
  title        = {Pith review of: A Unified Variational Framework for Optimal Transport with Lagrangian Costs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6NKC334}},
  note         = {Machine review of arXiv:2607.15471}
}

read the original abstract

We investigate optimal transport distances induced by general Lagrangian action functionals. Extending the classical Monge - Kantorovich and Benamou - Brenier theories, we derive a unified variational framework that connects several equivalent formulations of the induced transport distance, including Lagrangian, Eulerian, convex optimization, Hamilton - Jacobi dual, and Hamiltonian flow formulations. Under standard convexity assumptions on the Lagrangian, we establish the equivalence of these formulations through variational arguments and convex duality. The resulting optimality system reveals a natural Hamiltonian structure on the Wasserstein space, providing a direct link between optimal transport, Hamiltonian dynamics, and optimal control. The proposed framework extends the classical quadratic-cost theory to general Lagrangian costs and offers a unified perspective for the analysis of transport metrics generated by action functionals.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 1 linked inside Pith

[6]

Optimal mass transportation and Mather theory.J

Patrick Bernard and Boris Buffoni. Optimal mass transportation and Mather theory.J. European Math., Soc. 9: 85–121, 2007

2007
[19]

Fathi and A

A. Fathi and A. Figalli. Optimal Transportation on Non-compact Manifolds.Israel Journal of Mathematics, 175(1):1–59, 2010

2010
[37]

C. Villani. Optimal Transport: Old and New.Vol. 338, Springer Science & Business Media, 2008. EQUIV ALENT FORMULATIONS OF LAGRANGIAN OPTIMAL TRANSPORT 19 Iowa State University, Mathematics Department, Ames, IA 50011 Email address:hliu@iastate.edu

2008
[1]

Ambrosio and W

L. Ambrosio and W. Gangbo. Hamiltonian ODEs in the Wasserstein Space of Probability Measures. Communications on Pure and Applied Mathematics, 61(1), 18–53, 2008

2008
[2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savare. Gradient Flows in Metric Spaces and in the Space of Probability Measures.Lectures in Mathematics, Birkh ¨Aauser Verlag, Basel, 2005

2005
[3]

Wasserstein Generative Adversarial Networks

Arjovsky, Martin and Chintala, Soumith and Bottou, L´ eon. Wasserstein Generative Adversarial Networks. Proceedings of ICML, 214–223, 2017

2017
[4]

Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44:375–417, 1991

1991
[5]

J-D Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem.Numer. Math., 84: 375–393, 2000

2000

Show all 37 references

[7]

and Capuzzo-Dolcetta, I

Bardi, M. and Capuzzo-Dolcetta, I. Optimal Control and Viscosity Solutions of Hamil- ton–Jacobi–Bellman Equations. Birkh¨ auser, 1997

1997
[8]

Y. Brenier. Extended Monge-Kantorovich theory.Optimal transportation and applications (Martina Franca, Lecture Notes in Math, Springer, Berlin, Volume 1813: 91–121, 2003

2003
[9]

On the Global Convergence of Gradient Descent for Over- parameterized Models Using Optimal Transport

L´ ena ¨ ıc Chizat and Francis Bach. On the Global Convergence of Gradient Descent for Over- parameterized Models Using Optimal Transport. Advances in Neural Information Processing Sys- tems (NeurIPS 2018)

2018
[10]

Optimal Transport for Domain Adaptation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9), 1853–1867, 2017

Courty, Nicolas and Flamary, R´ emi and Tuia, Devis. Optimal Transport for Domain Adaptation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9), 1853–1867, 2017. 18 HAILIANG LIU

2017
[11]

Sinkhorn Distances: Lightspeed Computation of Optimal Transport

Marco Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal Transport. Advances in Neural Information Processing Systems, 26, 2292–2300, 2013

2013
[12]

Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David. Neural Ordinary Differential Equations. NeurIPS, 2018

2018
[13]

S.-N. Chow, W. Li, and H. Zhou. Wasserstein Hamiltonian flows. J. Di!erential Equations, 268 (2020), pp. 1205–1219

2020
[14]

Direct Methods in the Calculus of Variations

Dacorogna, Bernard. Direct Methods in the Calculus of Variations. Springer, 2nd edition, 2008

2008
[15]

Weinan, Jiequn Han, and Qianxiao Li

E. Weinan, Jiequn Han, and Qianxiao Li. A Mean-Field Optimal Control Formulation of Deep Learning. Research in Mathematical Sciences, 6(10), 2019

2019
[16]

Partial differential equations and Monge–Kantorovich mass transfer.Lecture Notes, 1989

Evans, L.C. Partial differential equations and Monge–Kantorovich mass transfer.Lecture Notes, 1989

1989
[17]

Evans, L. C. Partial Differential Equations. 2nd ed., American Mathematical Society, 2010

2010
[18]

POT: Python Optimal Transport

Flamary, R´ emi and Courty, Nicolas. POT: Python Optimal Transport. Journal of Machine Learning Research, 18(78), 1–8, 2017

2017
[20]

Figalli, W

A. Figalli, W. Gangbo, and T. Yolcu. A variational method for a class of parabolic PDEs.Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie, 10(1): 2011, 207–

2011
[21]

Weak KAM Theorem in Lagrangian Dynamics

Albert Fathi. Weak KAM Theorem in Lagrangian Dynamics. Cambridge University Press, 2019

2019
[22]

Gangbo and R

W. Gangbo and R. J. Mccann. The geometry of optimal transportation.Acta Math., 177:113–161, 1996

1996
[23]

Jordan, D

R. Jordan, D. Kinderlehrer, and F. Otto. The variational formulation of the Fokker–Planck equa- tion.SIAM J. Math. Anal., 29(1), 1–17, 1998

1998
[24]

Kantorovich

L.V. Kantorovich. On a problem of Monge.Uspekhi Mat. Nauk., 3: 225–226, 1948

1948
[25]

Knott and C.S

M. Knott and C.S. Smith. On the optimal mapping of distributions.J. Optim.Theory Appl., 43(1): 39–49, 1984

1984
[26]

S. Lisini. Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces.ESAIM Control Optim. Calc. Var., 15(3): 2009, 712–740

2009
[27]

Mean field games.Japanese Journal of Mathematics, 2 (1): 229–260, 2007

Lasry, Jean-Michel; Lions, Pierre-Louis. Mean field games.Japanese Journal of Mathematics, 2 (1): 229–260, 2007

2007
[28]

Liu and A

H. Liu and A. Tzavaras. Variational structure of Fokker-Planck equations with a variable matrix mobility. arXiv:2505.10676, 2025

2025 arXiv
[29]

R.J. McCann. A convexity principle for interacting gases.Adv. Math., 128(1):153–179, 1997

1997
[30]

A Mean Field View of the Landscape of Two-Layer Neural Networks

Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A Mean Field View of the Landscape of Two-Layer Neural Networks. Proceedings of the National Academy of Sciences (PNAS), 115(33): E7665–E7671, 2018

2018
[31]

F. Otto. The geometry of dissipative evolution equations: the porous medium equation.Comm. Partial Differential Equations, 26 (1-2), 101–174, 2001

2001
[32]

Computational Optimal Transport

Gabriel Peyr´ e and Marco Cuturi. Computational Optimal Transport. Foundations and Trends in Machine Learning, 11(5-6), p355-607, 2019

2019
[33]

Rachev and L

S.T. Rachev and L. R¨ uschendorf. Mass transportation problems. Vol. I and II. Probability and its Applications, Springer, New York, 1998

1998
[34]

The Earth Mover’s Distance as a Metric for Image Retrieval

Rubner, Yossi and Tomasi, Carlo and Guibas, Leonidas J. The Earth Mover’s Distance as a Metric for Image Retrieval. International Journal of Computer Vision, 40, 99–121, 1998

1998
[35]

Rotskoff and Eric Vanden-Eijnden, Neural Networks as Interacting Particle Systems: Asymptotic Convexity of the Loss Landscape and Universal Scaling of the Approximation Error

Grant M. Rotskoff and Eric Vanden-Eijnden, Neural Networks as Interacting Particle Systems: Asymptotic Convexity of the Loss Landscape and Universal Scaling of the Approximation Error. Communications on Pure and Applied Mathematics, 75(1): 188–248, 2022

2022
[36]

C. Villani. Topics in optimal transportation, Graduate Studies in Mathematics.Vol. 58, American Mathematical Society, Providence, R.I., 2003

2003

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [6]

Optimal mass transportation and Mather theory.J

Patrick Bernard and Boris Buffoni. Optimal mass transportation and Mather theory.J. European Math., Soc. 9: 85–121, 2007

2007

[2] [19]

Fathi and A

A. Fathi and A. Figalli. Optimal Transportation on Non-compact Manifolds.Israel Journal of Mathematics, 175(1):1–59, 2010

2010

[3] [37]

C. Villani. Optimal Transport: Old and New.Vol. 338, Springer Science & Business Media, 2008. EQUIV ALENT FORMULATIONS OF LAGRANGIAN OPTIMAL TRANSPORT 19 Iowa State University, Mathematics Department, Ames, IA 50011 Email address:hliu@iastate.edu

2008

[4] [1]

Ambrosio and W

L. Ambrosio and W. Gangbo. Hamiltonian ODEs in the Wasserstein Space of Probability Measures. Communications on Pure and Applied Mathematics, 61(1), 18–53, 2008

2008

[5] [2]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savare. Gradient Flows in Metric Spaces and in the Space of Probability Measures.Lectures in Mathematics, Birkh ¨Aauser Verlag, Basel, 2005

2005

[6] [3]

Wasserstein Generative Adversarial Networks

Arjovsky, Martin and Chintala, Soumith and Bottou, L´ eon. Wasserstein Generative Adversarial Networks. Proceedings of ICML, 214–223, 2017

2017

[7] [4]

Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44:375–417, 1991

1991

[8] [5]

J-D Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem.Numer. Math., 84: 375–393, 2000

2000