Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem

Alexandre Caboussat; Anna Peruso

REVIEW 4 major objections 5 minor 31 references

Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Input convex neural networks trained on the Monge–Ampère equation can approximate the optimal transport map as the gradient of a convex potential, with L2 errors around 1e-3 on test problems with known solutions.

desk verdict A promising PINN-OT idea whose central convexity claim is unsupported as written, because the paper never says how ICNN weights stay nonnegative during training. read the letter →

arxiv 2501.10162 v2 pith:4OMN6YVD submitted 2025-01-17 math.NA cs.LGcs.NA

Alexandre Caboussat , Anna Peruso This is my paper · ORCID

classification math.NAcs.LGcs.NA MSC 35J9649Q2265N9968T07

keywords optimaltransportMonge-Ampèreequationinputconvexneuralnetworksphysics-informedboundaryconditionsHausdorffdistancesensitivityanalysislogistics

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

The paper proposes a physics-informed neural network method for the quadratic-cost optimal transport problem in two and three dimensions. It claims that an input convex neural network can approximate the convex potential whose gradient is the optimal transport map, by minimizing a loss that combines a generalized Monge–Ampère equation residual $\det(D^2u)=f/g(\nabla u)$ with a discrete Hausdorff-distance boundary term enforcing $\nabla u(\partial X)=\partial Y$. On test problems with known solutions, the gradient of the trained network matches the exact transport map to errors around $10^{-3}$ in $L^2$ norm. If this holds, the method offers a mesh-free, architecture-enforced-convexity route to optimal transport maps, with the transport boundary condition handled without prescribing a boundary potential.

What carries the argument

The central object is the input convex neural network (ICNN), a feedforward network whose hidden-to-hidden weight matrices are constrained to have nonnegative entries and whose activation is convex and nondecreasing, so the network output is convex in its input. The loss is the sum of the squared Monge–Ampère residual at interior collocation points and a boundary term $E_{OT}$ defined by the two-sided discrete Hausdorff distance between $\nabla u_{NN}(\partial X)$ and $\partial Y$. The training protocol uses an identity-map initialization for $\nabla u$, then Adam followed by L-BFGS, with softplus activations and weight-squaring to maintain positivity.

What would settle it

Take a two-dimensional problem with a known nonlinear optimal map, run the protocol with a dense boundary discretization, and measure the one-sided Hausdorff distance from $\nabla u_{NN}(\partial X)$ to $\partial Y$; the paper's claim implies this distance decreases toward zero as the number of collocation points grows, so a persistent positive floor while the interior residual is small would falsify the boundary-condition enforcement.

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Extended reading notes

Core claim

The central claim is that the optimal transport map for quadratic cost between two absolutely continuous probability measures can be approximated by training an input convex neural network to solve the generalized Monge–Ampère equation $\det(D^2u)=f/g(\nabla u)$ with transport boundary condition $\nabla u(\partial X)=\partial Y$. Because the ICNN architecture guarantees convexity of the approximate potential $u_{NN}$ by construction, the trained gradient $\nabla u_{NN}$ is a legitimate candidate transport map. The paper demonstrates numerically on disk-to-ellipse, ellipse-to-rotated-ellipse, and Gaussian-to-uniform problems that this gradient converges to the exact map, and it enforces the transport boundary condition through a differentiable discrete Hausdorff-distance loss on the images of boundary collocation points, controlling both injectivity and surjectivity of the map.

Load-bearing premise

The load-bearing premise is that minimizing this nonconvex PINN loss from the identity-gradient initialization reliably lands on a network whose gradient approximates the exact optimal map, despite the authors' own observation that ICNNs are notoriously difficult to train and that run-to-run variability makes the error hard to control.

Editorial extensions

If this is right

The trained gradient $\nabla u_{NN}$ provides a direct approximation of the optimal transport map for quadratic cost, bypassing discrete linear-programming or iterative schemes.
The method handles non-constant source and target densities: Gaussian-to-Gaussian and bimodal-Gaussian-to-uniform examples show transported samples matching the target distribution, so it applies beyond uniform densities.
The Hausdorff-based boundary loss is generic: it requires only point clouds on $\partial X$ and $\partial Y$, so it can enforce transport boundary conditions for arbitrary target shapes, and simplifies to a Neumann condition for square-to-square maps.
The sensitivity experiments indicate that increasing the number of collocation points or training epochs decreases the gradient error, and that setting the number of boundary points equal to the number of collocation points is the best configuration.

Reading between the lines

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

Editorial inference: the identity-map initialization plus ICNN convexity could serve as a warm start for higher-dimensional or obstacle-constrained transport problems, where the authors expect training to scale poorly.
Editorial inference: because the Hausdorff boundary loss is defined on point sets, it could be extended to partial or semi-discrete boundary matchings without reformulating the PDE, a direction the paper does not pursue.
Editorial inference: a natural stress test is to compare the reported ~1e-3 gradient accuracy against a finite-difference or finite-element optimal transport solver on the non-analytic examples; the paper does not include such a comparison.

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Formalized claims in Lean

Claim #1: The central claim is that the optimal transport map for quadratic cost between two absolutely continuous probability measures can be approximated by training an input convex neural network to solve the generalized Monge–Ampère equation $\det(D^2u)=f/g(\nabla u)$ with transport boundary condition $\nabla u(\partial X)=\partial Y$. Because the ICNN architecture guarantees convexity of the approximat

/-- @claim 1 The central claim is that the optimal transport map for quadratic cost between two absolutely continuous probability measures can be approximated by training an input convex neural network to solve the generalized Monge–Ampère equation $\det(D^2u)=f/g(\nabla u)$ with transport boundary condition $\nabla u(\partial X)=\partial Y$. Because the ICNN architecture guarantees convexity of the approximat -/
def central_claim : Prop :=

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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

Colleague, here's my read of 2501.10162.

The paper has a genuinely useful idea: use an Input Convex Neural Network to solve the generalized Monge–Ampère equation with transport boundary conditions for quadratic-cost OT, and encode those boundary conditions as a discrete Hausdorff-distance loss. That specific combination does not appear in the cited Dirichlet-Monge–Ampère PINN work or the PICANN density-estimation work. The rotated-ellipse experiment, which has an explicit solution, shows pointwise errors around 1e-3 and the authors do a decent sensitivity analysis over random samples. The acknowledgement in Section 5 that the PINN error is 'rather difficult to control' is honest.

But there is a load-bearing gap. The ICNN architecture requires all W(l) to be nonnegative throughout training. The paper only says the weights are initialized by squaring them. It never says what happens during Adam/L-BFGS updates. Unconstrained gradient steps can push entries negative, and once that happens the network is no longer convex, so the gradient map is not necessarily the OT map. Clipping was tried and deemed unsatisfactory, which makes the missing mechanism even more pressing. I'd want to see either a genuine reparametrization (W = A^2 or softplus), a projection step, or at least a post-training check that the trained weights are nonnegative and the Hessian is PSD on the collocation points. Without that, the central claim of the paper is unverified.

There are smaller problems. The exact solution in Section 4.2 uses an offset of +7 in the first gradient component, but the ellipse is centered at x1 = 3.5, so for x1 in [−1,1] the gradient maps to [5,9] instead of [1.5,5.5]. That looks like a typo, but it undermines the reported validation numbers for that case. The Gaussian examples (4.4–4.7) provide only histograms and no quantitative error measures; the authors claim the error is 'of a similar order' without reporting values. And there are no baseline comparisons to standard OT solvers. The lack of public code is a minor annoyance; 'available on request' is fine but not ideal.

Overall the idea is worth pursuing and the paper deserves a serious referee, but as written it needs a substantive revision, not just copy-editing. The authors must clarify how convexity is maintained during optimization and correct the exact-solution mistake. If they can do that, the method would be a solid contribution to the PINN-OT literature.

Referee Report

4 major / 5 minor

Summary. The paper proposes a physics-informed neural network (PINN) method for the quadratic-cost optimal transport problem, writing the optimal map as the gradient of a convex potential u satisfying the generalized Monge-Ampère equation det(D^2u) = f/g(∇u) with the transport boundary condition ∇u(∂X) = ∂Y. The authors represent u by an input-convex neural network (ICNN), impose the PDE through a strong-form residual, and enforce the boundary condition through a discretized Hausdorff-distance loss. Numerical experiments cover disk-to-ellipse, ellipse-to-rotated-ellipse, Gaussian-to-uniform, Gaussian-to-Gaussian, bimodal Gaussian-to-uniform, and a 3D cube example; the Adam and L-BFGS optimizers are compared, and sensitivity to epochs, collocation points, and boundary-point ratio is studied. The claimed result is an accurate convex neural potential whose gradient approximates the optimal transport map, with reported L2 errors around 1e-3 in the exact-solution cases.

Significance. The combination of ICNNs with a PDE residual and a Hausdorff-type boundary loss is natural and potentially useful, and the paper includes honest sensitivity studies and an explicit exact-solution test in Section 4.3 that allows pointwise error evaluation. If the implementation issues identified below are resolved, the method could be a competitive tool for moderate-dimensional optimal transport problems. At present, however, the central claim that the architecture enforces convexity is not substantiated, because the positivity of the ICNN weights is only described at initialization. In addition, the main validation in Section 4.2 targets an incorrect exact solution, and the Gaussian experiments lack quantitative error reporting. The rotated-ellipse test gives credible pointwise errors around 1e-3, and the authors are transparent about run-to-run variability, but the missing details on convexity preservation and the Section 4.2 inconsistency are load-bearing and need to be fixed.

major comments (4)

[Section 3.2 and Section 4.1] The ICNN in Eq. (3.1) is convex only when all W(l) entries are nonnegative. Section 4.1 says the weights are initialized by element-wise squaring, but the subsequent training uses unconstrained Adam followed by L-BFGS (Table 1), and no projection, clipping, or reparametrization is described. Ordinary gradient steps can make entries of W(l) negative, in which case the trained network is not an ICNN and ∇u_NN is not guaranteed to be the gradient of a convex potential. Please state explicitly how W(l) ≥ 0 is maintained throughout training, or show empirically that it is preserved; without this, the assertion that convexity is enforced by construction is unsupported.
[Section 4.2] The exact map stated after the definition of the ellipse, ∇uex(x) = (2x1+7, (1/2)x2), is inconsistent with Y = { (y1-3.5)^2/2^2 + y2^2/0.5^2 < 1 }. For x1 ∈ [-1,1], 2x1+7 ranges over [5,9], i.e. an ellipse centered at 7, not at 3.5. The correct affine map for the stated Y is (2x1+3.5, 0.5x2). As written, Table 2 and Figures 2-4 report errors against a solution of a different problem; please correct either the definition of Y or the exact map and recompute the reported errors.
[Sections 3.3 and 4.2-4.7] The PDE residual is defined via the ratio f(x)/g(∇u(x)), with g(y) = (1/π)χ_Y(y) or g = χ_{[0,1]^2}. During training, ∇u_NN(x_i) can leave the support of g, making the denominator zero and the loss undefined at those collocation points. The manuscript does not describe any regularization, clamping, or reformulation that makes the residual finite in this situation. This is a genuine issue for compactly supported target densities and should be addressed, for example by using a smoothed density or by explicitly treating points outside the target support.
[Sections 4.4-4.7] The Gaussian and bimodal experiments are only evaluated visually, through histograms and pointwise error plots; no error norms, convergence rates, or quantitative comparisons are given. Section 4.4 states that the error 'is still of a similar order', but no numerical value supports that claim. Since these tests are the main evidence for nonuniform densities and for the 3D extension, at least a quantitative error measure (e.g., against a reference solution or a histogram-distance metric) should be reported.

minor comments (5)

[Figure 4 and Figure 12 captions] The captions label both bottom panels as 'Bottom left' and 'Top right' for the second component and its error; the second error panel should be 'Bottom right'.
[Section 3.3, Eq. (3.4)] Calling the first term of EOT a control of 'injectivity' is not accurate; the term ∑ dist(∇u(x_i), {y_j})^2 only enforces that each image lies near the target boundary, not that the map is injective. Please rephrase or justify.
[Section 4.3] The explicit optimal map for two centered ellipses, T(x)=M_Y R_θ M_X^{-1}x with the given formula for θ, is asserted without derivation or reference; a citation or a short argument would help the reader trust this exact solution.
[Section 5] The conclusion that the model 'is not particularly affected by the random sampling' is difficult to reconcile with Table 2 (L-BFGS L2 error 0.00267 ± 0.00376) and with the large sample variation visible in Figure 6; please qualify this statement.
[Section 4.1] The statement that the code is 'available on request' is not a reproducible-software statement; a persistent repository link would strengthen the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PINN loss is derived from the PDE and transport boundary condition, and exact solutions appear only in post-hoc validation.

full rationale

This paper's derivation chain is self-contained. The objective minimized is the PDE residual E_PDE, defined as the mismatch of det(D^2 u_NN) with f/g(∇u_NN), together with a discrete Hausdorff boundary loss E_OT that transcribes the transport boundary condition ∇u(∂X)=∂Y; both terms come directly from the mathematical problem (3.3) and Brenier's theorem, not from any fitted target. The exact solutions in Sections 4.2 and 4.3 and the histogram comparisons in Sections 4.4-4.7 are used only for post-hoc validation, and the sensitivity studies select hyperparameters such as Nc and Nb/Nc rather than fitting network parameters to the exact answer. The architectural reliance on ICNNs is supported by external references [2,21,29], with no self-citation chain involving the present authors, so no load-bearing step reduces to the paper's own inputs. The one notable gap is not circularity: Section 4.1 states that weights are initialized by squaring to guarantee positivity but does not specify how W(l)≥0 is maintained during the Adam/L-BFGS updates, leaving the trained network's convexity unverified. That is a correctness or reproducibility risk, not a reduction of the claimed result to its assumptions.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method rests on Brenier's theorem and standard PINN approximation assumptions. No new free physical parameters are introduced; the hand-chosen hyperparameters (C, N_b/N_c, architecture) are listed as free parameters. No new entities are posited.

free parameters (3)

Boundary loss weight C = 1
Default constant set to 1 in all experiments; no ablation study is reported for this weight, which balances PDE residual and boundary enforcement.
Boundary-to-collocation point ratio N_b/N_c = 1
Chosen based on the sensitivity experiment in Figure 6 (bottom); the error is reported to be smallest at ratio 1, but this is an empirical selection, not a derived optimum.
Network architecture (hidden layers x neurons) = 4 x 10
Adopted following recommendations in [21,29]; no convergence study with respect to network size is provided.

assumptions (4)

domain assumption Brenier's theorem: existence and uniqueness of a convex potential u such that the optimal map is gradient u and satisfies det(D^2 u) = f/g.
Invoked in Section 2, based on [14]; requires absolutely continuous probability measures with densities bounded away from zero and infinity.
domain assumption Domains X and Y are bounded, smooth open sets in R^2 with strictly positive, bounded densities.
Assumed in Section 2; smoothness ensures regularity of the solution and well-posedness of the boundary condition.
standard math Universal approximation theorem for feedforward networks.
Cited in Section 3.1 to justify that neural networks can approximate the solution; no rate or dimension-dependent guarantee is used.
ad hoc to paper The discrete Hausdorff loss with finite boundary point sets is a valid proxy for the continuous transport boundary condition gradient u(partial X) = partial Y.
Introduced in Section 3.3, Eq (3.4); the paper relies on this approximation for all numerical results but provides no error analysis linking the discretization to the continuous condition.

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Cite this review

Pith. "Pith review of Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem." pith.science (2026). https://pith.science/paper/4OMN6YVD

@misc{pith2026250110162,
  author       = {Pith},
  title        = {Pith review of: Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OMN6YVD}},
  note         = {Machine review of arXiv:2501.10162}
}

read the original abstract

Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge-Amp`ere equation. Convex neural networks are advocated to enforce the convexity of the solution to the Monge-Amp\`ere equation and obtain a suitable approximation of the optimal transport map. A particular focus is set on the enforcement of transport boundary conditions in the loss function. Numerical experiments illustrate the solution to the optimal transport problem in several configurations, and sensitivity analyses are performed.

Figures

Figures reproduced from arXiv: 2501.10162 by the authors.

**Figure 1.** Sketch of the optimal transport problem, illustrating the mapping of a domain X onto another domain Y. Monge problem is ill-posed due to the non-existence of a transport map; for instance, when µ is a Dirac measure and ν is not. This suggests that certain conditions on the measures µ and ν are necessary. Consider the case where X , Y ⊂ R n, with µ(dx) = f(x) dx and ν(dy) = g(y) dy. If s : X → Y is a sufficiently s… view at source ↗

**Figure 2.** Transport of a constant density from a unit disk domain to an ellipse domain. Visualization of uniformly sampled source points and their image through exact and approximate transport maps [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Transport of a constant density from a unit disk domain to an ellipse domain. Visualization of the transport map ∇uNN [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: Transport of a constant density from a unit disk domain to an ellipse domain. Top left: first x-coordinate of the transport map ∇uNN ; Top right: Pointwise absolute error |∇uNN,x − ∇uex,x|; Bottom left: second y-coordinate of the transport map ∇uNN ; Top right: Pointw…

**Figure 5.** Figure 5: Transport of a constant density from a unit disk domain to an ellipse domain. Left: Loss function vs epochs. Right: ∥∇u − ∇uNN ∥L2 vs epochs, with the error computed on a validation set. For the L-BFGS optimizer, the total epoch count is taken as the number of (outer…

**Figure 6.** Figure 6: Transport of a constant density from a unit disk domain to an ellipse domain. Top: L 2 (test) error ||∇uex − ∇uNN ||L2(X ) vs number of epochs. Middle: L 2 (test) error ||∇uex − ∇uNN ||L2(X ) vs number of collocation points. Bottom: L 2 (test) error ||∇uex − ∇uNN ||L2…

**Figure 7.** Figure 7: Transport of a constant density from an ellipse domain to a rotated ellipse domain. Visualization of uniformly sampled source points and their image through exact and approximate transport maps [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Transport of a constant density from an ellipse domain to a rotated ellipse domain. Visualization of the transport map ∇uNN [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Transport of a constant density from an ellipse domain to a rotated ellipse domain. Top left: first x-coordinate of the transport map ∇uNN ; Top right: Pointwise absolute error |∇uNN,x − ∇uex,x|; Bottom left: second y-coordinate of the transport map ∇uNN ; Top right: …

**Figure 10.** Figure 10: Transport of a constant density from an ellipse domain to a rotated ellipse domain. Loss function vs epochs. For the L-BFGS optimizer, the total epoch count is taken as the number of (outer) L-BFGS iterations times the number of inner subiterations. Shaded bands sh…

**Figure 11.** Figure 11: illustrates the approximated optimal transport map. The histogram in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: Transport of a Gaussian density to a uniform density. Top left: first x-coordinate of the transport map ∇uNN ; Top right: Pointwise absolute error |∇uNN,x − ∇uex,x|; Bottom left: second y-coordinate of the transport map ∇uNN ; Top right: Pointwise absolute error |∇uNN…

**Figure 13.** Figure 13: Transport of a Gaussian density to another Gaussian density. Left: sampling of points according to original Gaussian density; Right: transported solution with approximated solution ∇uNN . (N = 100000 points sampled into the cells of a structured grid of size 80 × 80).…

**Figure 14.** Figure 14: Transport of a bimodal Gaussian density to a uniform density. Left: sampling of points according to bimodal Gaussian density (4.2); Right: transported solution with approximated solution ∇uNN . (N = 100000 points sampled into the cells of a structured grid of size 80…

**Figure 15.** Figure 15: illustrates the approximated optimal transport map. We can observe that the transported solution re-dispatches appropriately uniformly the points on the grid [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

**Figure 16.** Figure 16: Transport of a 3D Gaussian density to a uniform density. Left: sample of points on the boundary of the unit cube ∂X , uniformly sampled on the boundary Right: location of the boundary points transported with the approximated solution ∇uNN . The relatively high compu…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Ambrosio, L

L. Ambrosio, L. A. Caffarelli, Y. Brenier, G. Buttazzo, C. Villani, and S. Salsa. Optimal Transportation and Applications. Springer Berlin, Heidelberg, 2003

work page 2003

[2] [2]

B. Amos, L. Xu, and J. Z. Kolter. Input convex neural networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learn- ing, volume 70 of Proceedings of Machine Learning Research, pages 146–155. PMLR, 06–11 Aug 2017

work page 2017

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