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REVIEW 2 major objections 5 minor 24 references

Wasserstein Residuals: Learning Gradient Flows from Population Dynamics

T0 review · 2 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A residual loss that vanishes exactly on Wasserstein gradient flows yields a simulation-free particle method that recovers population trajectories from sparse snapshots better than JKO schemes.

desk verdict Solid residual framing plus a practical simulation-free particle method that actually beats JKO baselines on EB and unpaired recovery; the soft residual/KDE gap is real but already owned by the authors and does not sink the empirical case. read the letter →

arxiv 2607.04738 v1 pith:XJTO3ZWX submitted 2026-07-06 stat.ML cs.AIcs.LG

Markus Heinonen , Yair Shenfeld , Ricardo Baptista , Daniel Waxman , Dmitry Batenkov , Tim Cooijmans , Eli Bingham This is my paper · ORCID
classification stat.MLcs.AIcs.LG MSC 49Q2262G0735Q84
keywords WassersteingradientflowpopulationdynamicstrajectoryinferencevelocityresidualJKOschemeparticlemethodsingle-cellRNAinteractionkernels
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 argues that recovering an energy whose Wasserstein gradient flow fits observed population snapshots should not be forced through the Jordan–Kinderlehrer–Otto proximal scheme. Instead, a nonnegative velocity residual that is zero if and only if the curve is a gradient flow of the energy, added to a data-fitting divergence, produces a single global objective. This residual view unifies earlier path-finding and action-matching ideas and immediately suggests stitching: both the energy and a kernel-density particle trajectory are learned jointly, with no ODE simulation and no optimal-transport couplings between consecutive snapshots. Because the trajectory is an explicit first-class variable, large temporal gaps no longer force straight-line chords. On standard single-cell trajectory benchmarks the method reports the lowest distributional errors among published baselines.

What carries the argument

The velocity residual R_vel[F,(ρ,v)] = ∫‖v_t + ∇(δF/δρ_t)‖² ρ_t dx dt, which is zero precisely when ρ is a Wasserstein gradient flow of F; stitching realizes it by a learnable KDE particle cloud whose velocities are the particle derivatives themselves.

What would settle it

On a synthetic landscape with known curved trajectories and deliberately large observation gaps, check whether the learned residual remains near zero while the recovered potential and particle paths still match ground truth; systematic residual floor or collapsed potentials would refute the claim.

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Extended reading notes

Core claim

Minimizing a velocity residual that enforces the continuity equation together with a data divergence recovers both the energy functional and the full continuous trajectory of a Wasserstein gradient flow from discrete, possibly unpaired population snapshots; the resulting particle method (stitching) is simulation-free, tolerates large observation gaps, and attains state-of-the-art accuracy on trajectory-inference tasks.

Load-bearing premise

That approximating the density by its particle centers and softly penalizing a nonzero residual is enough to recover a true gradient flow, even though the residual is never forced exactly to zero and infinite-particle convergence is unproved.

Editorial extensions

If this is right

  • Trajectory inference from sparse single-cell or crowd snapshots can be performed without solving optimal-transport problems at every time step.
  • Both potential and interaction kernels can be recovered jointly from marginals alone, including phase transitions that form clusters or rings.
  • The same residual objective extends immediately to non-gradient (chiral or non-reciprocal) velocity fields by replacing the Wasserstein gradient with a free vector field.
  • Time discretization of the trajectory becomes independent of the observation schedule, so irregularly sampled data no longer force first-order chords.

Reading between the lines

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

  • If the residual view is correct, any continuity-equation-based dynamics (not only pure gradient flows) becomes learnable by the same particle stitching construction, opening a route to transformers and active-matter models from snapshots alone.
  • The O(N²) entropy and interaction cost may be the practical bottleneck long before residual bias; sparse or low-rank kernel approximations would be a natural next algorithmic step.
  • Because the trajectory is free, the method could be used as a diagnostic: large residual after training would flag that the data are inconsistent with any energy of the assumed form.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a residual framework for learning Wasserstein gradient flows (WGFs) from sparse population snapshots. Instead of the dominant JKO proximal scheme, it enforces the continuity equation via nonnegative density or velocity residuals whose zero set is exactly a WGF of a functional F, then couples this residual to a data-fitting divergence into a single global objective. This perspective is used to unify Path-Finding and Action Matching, and to introduce stitching: a simulation-free particle method that jointly learns F (potential, entropy, interaction) and a KDE-parametrized curve of trajectories. Empirically, stitching is shown to track curved trajectories under large observation gaps, recover synthetic potentials under paired and unpaired snapshots, achieve state-of-the-art W1/W2 on the embryoid-body single-cell benchmark (full-data and leave-two-out), disentangle potential and interaction kernels, and extend to non-gradient chiral dynamics.

Significance. If the claims hold, the residual view is a useful organizing principle for inverse WGF problems and stitching is a practical alternative to JKO-based methods: it avoids repeated OT solves, decouples temporal discretization from observation times, and is robust to unpaired snapshots and large gaps. The EB results (mean W1 about 0.54–0.58 full-data; mean W2 about 0.88 leave-two-out) and the interaction-recovery experiment are competitive with strong published baselines. Lemma 1 correctly links the velocity residual to the EDE form under stated regularity; Claims 1–2 justify the KDE velocity and centers limit. Public code is a clear strength. The main scientific value is algorithmic and empirical rather than a new existence theory.

major comments (2)
  1. [§7, Tables 2–3, Eq. (6)/(13)] Section 7 Limitations and the stitching objective (Eq. 6, 10, 13): the residual is a soft regularizer and does not force R_vel = 0. On synthetic data, L2-UVP and pattern R2 partially validate that recovered gradients match ground truth. On the EB benchmark (Tables 2 and 5), only data-fit W1/W2 are reported. Without post-training residual magnitudes (or a held-out check that learned velocity is close to -grad delta F / delta rho), the interpretation that the SOTA numbers correspond to a recovered WGF—rather than a flexible trajectory fit—is not fully supported. Please report residual scale after training for the main EB and interaction runs, and ideally a simple residual-vs-data-weight ablation.
  2. [§4, Eq. (11)–(13), Appendix C] Empirical centers approximation (Eq. 11–13) and Appendix C: the method replaces the KDE measure by Dirac centers and uses leave-one-out score evaluation to avoid self-kernel collapse. This is load-bearing for inverse recovery, yet bias relative to full KDE or KDE-MC quadrature (alpha) is not quantified on the main benchmarks. A short diagnostic (e.g., residual and L2-UVP vs alpha and bandwidth on one synthetic potential and one EB seed) would make the approximation’s adequacy checkable rather than assumed.
minor comments (5)
  1. [Figure 1, §5.1] Figure 1 caption and §5.1: R2 = 0.62 vs 0.51 is useful; please state whether this is raw or pattern R2 and on what support, for consistency with Appendix F.
  2. [Table 1] Table 1: several L2-UVP entries for JKO methods are large or negative-looking (e.g., 149, -0.00). Clarify units/normalization and why some cells are marked with dashes for iJKOnet; a short note in the caption would help.
  3. [§3–4] Notation: rho^theta vs rho_t, and F_theta vs F, switch between continuous and discrete forms; a brief notation paragraph at the start of §4 would reduce friction.
  4. [§6, Appendix D] Related work: GenWGP (Liu and Zhou) and Action Matching are well placed in Appendix D; a one-sentence contrast in the main §6 on simulation cost vs KDE approximation would help readers who skip the appendix.
  5. [Throughout] Typos/style: “side-steps” → “sidesteps” (Appendix D.2); “Bd2_W2-UVP” formatting is inconsistent across tables; arXiv id and “Preprint” header are fine for review but should be cleaned for journal submission.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: residual objective plus external data divergences and held-out metrics; SOTA claims are empirical, not algebraic restatements of fitted inputs.

full rationale

The paper defines a nonnegative velocity residual R_vel that vanishes iff the curve is a WGF of F (Section 3, Lemma 1), then adds an independent data-fitting divergence D(ρ_t, q_t) at observed times to form the global objective (Eq. 6 / stitching objective Eq. 13). The residual is a soft regularizer, not a hard constraint that forces R=0 by construction; the paper itself states this limitation (§7). Curve and functional are co-parametrized and optimized jointly, but evaluation uses external benchmarks (EMD, L2-UVP, Bd2_W2-UVP on synthetic potentials; W1/W2 on EB full-data and leave-two-out splits) that are not algebraic rearrangements of the training loss. Pattern R2 is scale-invariant because WGF dynamics are only identifiable up to scale (domain structure, not circular fitting). Concurrent residual work (Liu & Zhou GenWGP; Neklyudov Action Matching) is cited as related instantiations of the same residual idea and is disclosed as concurrent/related, not as a load-bearing uniqueness theorem that forbids alternatives. No fitted scalar is renamed as a prediction; no self-citation chain forces the SOTA numbers. Minor residual/KDE approximation concerns are correctness/limitation issues, not circularity. Score 1 for ordinary related-work self-reference that is not load-bearing.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The central claim rests on classical WGF theory (continuity equation, first variation of potential+entropy+interaction energies), standard divergences for data fit, and several modeling/approximation choices (KDE centers, soft residual weight, neural V/W). Free parameters are the usual ML knobs; no new physical particle is postulated. Invented entities are methodological (residuals taxonomy, stitching).

free parameters (5)
  • residual weight λ / w_vel vs data-fit weight
    Balances velocity residual against D(ρ_t,q_t); chosen per experiment, not derived.
  • particle count N and trajectory nodes K
    Capacity of the learnable curve (e.g. N=1000/K=50 synthetic; N=100/K=50 EB); hand-set.
  • KDE bandwidth and mixture weights
    Initialized (Silverman/softplus) and often trained; controls density and score estimates.
  • MLP widths and coefficients c_V, c_H, c_W
    Architecture (typically 64,64) and softplus energy scales fitted during training.
  • time-discretization scheme and α centers/MC mix
    Trapezoidal default, α=0 or 0.5; affects residual bias/variance.
assumptions (5)
  • standard math Absolutely continuous curves in (P2,W2) admit a unique minimal velocity field satisfying the continuity equation (Ambrosio/Santambrogio).
    Background for density and velocity residuals (§2, App. A).
  • standard math Tangent, JKO, EVI, and EDE characterizations of metric gradient flows are equivalent under suitable regularity (Ambrosio et al. Ch. 11).
    Justifies focusing on tangent/EDE residuals rather than EVI/JKO (§3).
  • domain assumption True energy is of form potential + entropy + pairwise interaction with analytic Wasserstein gradient (Eq. 4–5).
    Restricts F_θ parametrization; stated as focus of the work (§2, Goal).
  • ad hoc to paper Empirical centers approximation of the KDE residual (Eq. 11–13) is accurate enough for inverse recovery.
    Core computational reduction of stitching; exact only as ϕ→δ (App. C).
  • domain assumption Observed snapshots are generated by (or well-approximated by) a WGF of such an F, possibly with small diffusion.
    Inverse problem goal; non-gradient extension is only demonstrated, not the main theory.
invented entities (2)
  • Wasserstein residual framework (density/velocity residuals + data divergence)
    purpose: Single global objective whose zero residual marks a WGF and unifies prior methods.
    Methodological construct; independent evidence is empirical residual near zero and benchmark wins, not an external physical prediction.
  • Stitching algorithm (learnable particle trajectories + KDE + residual loss) independent evidence
    purpose: Simulation-free, gap-robust inverse solver for F and ρ jointly.
    New algorithm; falsifiable via held-out marginal metrics and potential recovery R², which the paper reports.

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Cite this review

Pith. "Pith review of Wasserstein Residuals: Learning Gradient Flows from Population Dynamics." pith.science (2026). https://pith.science/paper/XJTO3ZWX

@misc{pith2026260704738,
  author       = {Pith},
  title        = {Pith review of: Wasserstein Residuals: Learning Gradient Flows from Population Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJTO3ZWX}},
  note         = {Machine review of arXiv:2607.04738}
}
read the original abstract

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

Figures

Figures reproduced from arXiv: 2607.04738 by the authors.

Figure 1
Figure 1. Stitching recovers curvature that JKO chord predictors miss. Sparse SDE snapshots (N = 20) on a sinusoidal valley at t ∈ {0, 5, 10, 20, 30}. (a) True potential. (b) Stitching’s trajectory tracks the valley. (c) Lightspeed (Terpin et al., 2024) does not capture the curved trajectories of the particles; see Section 5.1. • We recast the problem as minimizing a nonnegative residual term, a framework subsuming Path-Findi… view at source ↗
Figure 2
Figure 2. Stitching’s potential recovery is qualitatively unchanged when consecutive snapshots are decorrelated. Top: ground-truth V . Middle / bottom: stitching’s V θ trained on paired / unpaired snapshots. Per-panel labels: R2 pattern. Full 15-potential galleries in Appendix F. (EMD) in the paired regime, but degrade or collapse on the unpaired UVPs — JKOnet⋆ V in particular fails to recover an informative gradient field at… view at source ↗
Figure 3
Figure 3. Both static and time-varying V θ drive the population evolution accurately. Contours of V θ in (PC1, PC2) with PC3–5 fixed at the global mean; half-integer columns are unseen at training. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Stitching captures the Gaussian → ring → cluster phase transition and recovers V, W at pattern R2=0.84, 0.80. Top: observed data at eight times; rightmost cell shows learned V θ (x) filled, with V contours dashed. Bottom: Stitching KDE samples; rightmost cell shows Wθ …
Figure 5
Figure 5. Figure 5: Stitching tracks the chiral orbiting dynamics. Samples from the learned curve ρ θ (model) and the data q (observed) at particular time snapshots over the training window. 6 Related work In this section, we discuss some related approaches in learning potential functions…
Figure 6
Figure 6. Figure 6: Paired vs. unpaired snapshot evolution on the synthetic potentials following Persiianov et al. (2026). Top: In ‘paired’ setting we observe same particles driven by a potential gradient over time. Bottom: In ‘unpaired‘ trajectory structure is lost, as if the observation…
Figure 7
Figure 7. Figure 7: Stitching recovers the level-set geometry of V on 11 of 14 informative landscapes. Synthetic potential recovery, paired regime: true V (left) vs learned V θ (right) for each potential, mean-centered with shared per-potential color scale; titles report scale-invariant R…
Figure 8
Figure 8. Figure 8: Stitching remains robust to independently-sampled snapshots; detectable degradation only on the four most delicately structured potentials. Synthetic potential recovery, unpaired regime; same conventions as [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

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Reference graph

Works this paper leans on

24 extracted references · 2 linked inside Pith

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    12 A Mathematical background: Wasserstein gradient flows This appendix collects standard definitions and results for Wasserstein gradient flows used in the main text. We refer the reader to Ambrosio et al. (2005); Villani (2016); Santambrogio (2015) for comprehensive treatment. A.1 The Wasserstein space The space of probability measures with finite second...

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    who used this formulation to define gradient flows in Wasserstein space. Evolution Variational Inequality (EVI).The EVI definition for (xt)t∈[0,T] being a gradient flow offrequires the existence ofα∈Rsuch that 1 2 d dt ∥xt −y∥ 2 ≤f(y)−f(x t)− α 2 ∥xt −y∥ 2 for ally∈R D.(EVI E) For smooth α-convex f, the convexity inequality ∇f(x t)·(y−x t)≤f(y)−f(x t)− α ...

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    The first inequality is an equality if and only if ˙xr =−c r ∇f(x r) for some cr ≥0 , while the second inequality is an equality if and only if ∥˙xr∥=∥∇f(x r)∥

    = Z T 0 ∇f(x r)·˙x r dr≥ − Z T 0 ∥∇f(x r)∥ · ∥˙xr∥dr ≥ − Z T 0 1 2 ∥∇f(x r)∥2 + 1 2 ∥˙xr∥2 dr. The first inequality is an equality if and only if ˙xr =−c r ∇f(x r) for some cr ≥0 , while the second inequality is an equality if and only if ∥˙xr∥=∥∇f(x r)∥. Together (when ∇f(x r)̸= 0 ) the two equalities force cr = 1, i.e. ˙xt =−∇f(x t). It follows that (ED...

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    (26) Proof

    = Z T 0 Z RD δF δρt (x)∂ tρt(x) dxdt.(25) Then, Rvel[F,(ρ,v)] = Z T 0 Z RD vt(x) +∇ x δF δρt (x) 2 ρt(x) dxdt = 2 (F[ρT ]− F[ρ 0]) + Z T 0 " |˙ρt|2 W2 + Z RD ∇x δF δρt (x) 2 ρt(x) dx # dt. (26) Proof. By Ambrosio et al. (2005, Theorem 8.3.1), |˙ρt|2 W2 = R RD ∥vt(x)∥2 ρt(x) dx. Expanding the squared norm inR vel[F,(ρ,v)], and integrating by parts, Rvel[F,...

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    The entropy contribution then drops out of the velocity residual and particles collapse onto minima ofV θ instead of spreading. We drop particlejfrom both numerator and denominator when evaluating the score atx θ t,j: ∇x logρ θ t (xθ t,j)≈ P k̸=j wθ k ∇xϕ(xθ t,j −x θ t,k)P l̸=...

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    JKONet uses Brenier’s theorem to characterize the solution to (JKO) as an optimization problem over a convex function ψ

    iterates (JKO), starting from ρ0 =q 0, by solving a bi-level optimization. JKONet uses Brenier’s theorem to characterize the solution to (JKO) as an optimization problem over a convex function ψ. In particular, given a parametrized functional F ξ, the inner optimization proble...

    2021

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    In this style of algorithms the optimal transport problem is first solved between any two consecutive marginals in {qt}t∈Tobs

    algorithms use the optimality condition of (JKO), x′ −x h =−∇ δF δρt+h (x′)∀(x,x ′)∈suppγ t,(JKO-FO) where γt is the optimal transport plan between consecutive marginals in {qt}t∈Tobs. In this style of algorithms the optimal transport problem is first solved between any two co...

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    and flow matching (Lipman et al., 2023; Tong et al., 2024; Albergo and Vanden-Eijnden, 2023; Wang et al.,

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    They are thus solving a strictly weaker problem: they recover dynamics, but not an underlying energy

    learn velocity fields between distributions without imposing a gradient-flow structure. They are thus solving a strictly weaker problem: they recover dynamics, but not an underlying energy. Similarly to flow matching, normalizing flows parametrize a velocity network vθ t and l...

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    obtained by transporting ρ0 along vθ t . This provides exact normalized density evaluation, and permits learning the model parameters by maximizing the likelihood (i.e., minimizing the KL divergence) of the data at observed times under the modelρ θ t . D.6 Other methods We als...

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    Carrillo et al

    study identifiability of the drift and diffusion of an SDE from temporal marginals, complementary to our setting where the drift is constrained to be the gradient of a learned functional. Carrillo et al. (2025) considers recovery of interaction kernel from gridded density data...

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    Other hyperparameters: (64,64) hidden, lr= 10 −3, 10K iterations, OT-Hungarian coupling between consecutive snapshot pairs

    with one modification: the entropy coefficient is trainable rather than frozen at zero, matching stitching for fairness. Other hyperparameters: (64,64) hidden, lr= 10 −3, 10K iterations, OT-Hungarian coupling between consecutive snapshot pairs. Metrics.We report (i) per-snapsh...

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    R2 raw is sensitive to additive and multiplicative shifts of V θ

    report two coefficients of determination between the learned and ground-truth potentials, both evaluated on a uniform grid covering the data support.R 2 raw is the standard coefficient of determination, R2 raw(V θ, V) := 1− P i V θ(xi)−V(x i) 2 P i V(x i)− ¯V 2 , where ¯V is t...

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    (2026, Tab

    Because of compute, the baseline runs are restricted to the 6 potentials Persiianov et al. (2026, Tab

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    (2024), in both regimes

    Per-potential numbers.Table 4 reports stitching’s three metrics on each of the 15 landscapes of Terpin et al. (2024), in both regimes. The main-text Table 1 compares stitching against the JKO baselines on the 6 sensitive potentials. The consistency claim of the main text is vi...

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    25 G Single-cell trajectory inference: details Dataset and preprocessing.The embryoid body (EB) dataset of Moon et al. (2019) comprises ∼17,000 cells observed across five 3-day windows of human embryonic stem cell differentiation. We use the PCA-reduced version distributed by ...

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    and JKOnet∗ t,V in Terpin et al. (2024). • Trajectory: K= 50 learnable snapshots between tmin and tmax, N= 100 particles per snapshot, OT-coupled at initialization (Hungarian assignment + linear interpolation between consecutive observed marginals). • Density model:per-dimensi...

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    Evaluation protocols

    •Wall time:∼2minutes per run on a CPU laptop (M-series). Evaluation protocols. • Full data(Table 2): train on all five marginals; report W1 between the stitching KDE marginal at each observed t and the held-out test cells at that t. Forward-rollout from the learned trajectory;...

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    Pattern R2 for V and W .WGF inference from snapshots is identifiable only up to a joint (V, W, σ)- scale ambiguity (Persiianov et al., 2026, App

    on the same train/test split. Pattern R2 for V and W .WGF inference from snapshots is identifiable only up to a joint (V, W, σ)- scale ambiguity (Persiianov et al., 2026, App. A): only ∇V /σ and ∇W/σ are determined by the data, not the absolute scale of V, W . We therefore rep...

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    and the broader class of non-reciprocal collective systems (Fruchart et al., 2021), none of which arise as Wasserstein gradient flows. Concretely, for any model curveρ θ with Eulerian velocityv θ and any model velocity fieldu θ, Rvel[uθ,(ρ θ,v θ)] := Z T 0 Z RD vθ t (x)−u θ t ...

    2021

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