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A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs

Geuntaek Seo, Cheolhyeong Kim, Hwijae Son, Hyung Ju Hwang

arXiv:2610.07602v1math.NAcs.LGcs.NAmath.APmath.OC

TL;DR

Advection–reaction–diffusion systems combine spatial transport with local mass creation or loss, while computing HK-JKO steps requires minimizing an objective involving the next measure and unbalanced transport cost. The scheme updates measures with a spatial map and growth factor, using a feasible primal cone action to upper-bound the exact proximal cost. Under regular-pair approximation, parametric pairs recover exact-step densities and objectives, while under exact-step regularity, positive-semidefinite interactions, and global dual feasibility, a primal–dual gap controls suboptimality and Boltzmann-entropy L1 error.

  • Problem

    Advection–reaction–diffusion systems combine spatial transport with local mass creation or loss, while computing HK-JKO steps requires minimizing an objective involving the next measure and unbalanced transport cost.

    Source: §1.1 · pp. 1–2

  • Method

    The scheme updates measures with a spatial map and growth factor, using a feasible primal cone action to upper-bound the exact proximal cost.

    Source: §7 · p. 28

  • Results

    Under regular-pair approximation, parametric pairs recover exact-step densities and objectives, while under exact-step regularity, positive-semidefinite interactions, and global dual feasibility, a primal–dual gap controls suboptimality and Boltzmann-entropy L1 error.

    Source: §7 · p. 29

  • Takeaways & Limitations

    The analysis supplies time-discrete guarantees for exact HK-JKO steps and admissible computed pairs, including conditional recovery and a posteriori error estimates.

    Source: §7 · pp. 28–29

    The dissipation criterion is exact only when relevant functional values have controlled quadrature error, and energy decay alone does not rule out a nontrivial pure-reaction update.

    Source: §3.2 · p. 12

Abstract

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We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK) geometry of unbalanced optimal transport. Each update is parametrized by a spatial map and a mass-changing factor, allowing spatial redistribution and local mass creation or loss to be treated jointly within a single variational step. Their cone action bounds the squared HK distance from above, yielding a sufficient condition for discrete energy dissipation through comparison with the identity pair. Minimizing the pair objective over all admissible pairs recovers the exact JKO minimum when the source and a minimizer have positive densities. We establish existence and mass bounds for JKO minimizers and, under additional assumptions, obtain positivity and regularity together with a discrete Euler-Lagrange equation and a metric-dissipation identity. The self-consistent chemical potential is then nonincreasing along an optimal map. There exist parametric pairs whose endpoint densities and objective values converge to those of an exact JKO minimizer, provided a regular-pair approximation hypothesis holds. Finally, we show that a primal-dual gap controls objective suboptimality and, for Boltzmann entropy, the $L^1$ density error, assuming exact-step regularity, positive-semidefinite interactions, and global dual feasibility. Numerical experiments examine pointwise agreement with the PDE, energy dissipation, and the roles of transport, reaction, and fully implicit interactions.

1 Introduction

HK geometry couples spatial transport with local mass creation or loss, making it suitable for advection–reaction–diffusion flows; computing its JKO steps remains challenging. The paper introduces a Monge-growth primal formulation and analyzes its exact target, parametric approximation, computed-pair certificates, relation to prior work, and scope.

1.1 Hellinger–Kantorovich gradient flows and the computational problem

  • HK geometry combines spatial transport and local mass change, while grid-based methods scale poorly with dimension and motivate mesh-free parametric approaches.The paper studies gradient flows of free energies with internal, external, and nonlocal interaction terms.Source: §1.1 · pp. 1–2

1.2 The primal formulation

  • A Monge-growth pair jointly represents an update through a spatial map and mass-changing factor, with a primal cone action that upper-bounds the exact proximal cost.Under positive-density hypotheses, unrestricted pair optimization is equivalent to the exact JKO problem; comparison with the identity pair gives a descent test.Source: §1.1 · p. 2, §1.2 · pp. 2–3

1.3 Main results and contributions

  • The exact-step analysis establishes existence and mass control, with additional assumptions yielding positivity, density bounds, an Euler–Lagrange equation, and metric–dissipation identity.For Lipschitz external and interaction potentials, the new density is Lipschitz; the analysis includes fully implicit interactions.Source: §1.3 · p. 4
  • Under a regular-pair approximation hypothesis, parametric pairs converge in endpoint density and objective values to the exact step, with no rate uniform in the number of steps proved.At fixed time step, the result extends to a fixed finite horizon when the corresponding approximation hypothesis holds at each step.Source: §1.3 · p. 4
  • With positive-semidefinite interactions and global dual feasibility, a primal–dual gap controls objective suboptimality and, for Boltzmann entropy, the L1 density error.This a posteriori estimate does not require a parametric approximation hypothesis.Source: §1.3 · p. 4

1.4 Relation to previous work

  • Prior work includes HK minimizing movements, Monge and regularity theory, and neural transport solvers; this paper applies a primal cone action to fully implicit HK-JKO steps.The cited antecedents include direct HK minimizing movements [9], Monge properties, and neural methods [14] [20] [21].Source: §1.4 · p. 5
  • The restarted formulation applies a fresh pair to the current measure at each step, trading direct density evaluation from a fixed base law for propagated densities and growth-weighted particle updates.Recursive numerical runs additionally use a mass-corrected density fit after each pair update.Source: §1.4 · p. 5, §1.5 · p. 5

1.5 Scope and organization

  • The analysis covers time-discrete exact steps, pair formulations, and guarantees for admissible computed pairs, but proves neither stochastic-optimizer convergence nor a τ ↓ 0 limit for the trained scheme.The approximation and a posteriori results also carry their stated assumptions and limitations.Source: §1.5 · p. 5

2 Setting and the exact HK-JKO step

The exact HK-JKO step is posed for measures on bounded convex domains or flat tori, with transport and reaction represented jointly in HK geometry. Under the stated assumptions, minimizers exist, admit optimal Monge-growth representations, and have further density and dissipation properties under additional hypotheses.

2.1 The minimal setting

  • The minimal setting uses a bounded convex domain or flat torus, with Boltzmann entropy as a principal internal-energy example and assumptions on potentials and coercivity specified by theorem.The internal energy assumptions include strict convexity and U′(0+) = −∞ where the pointwise regularity setting is invoked.Source: §2.1 · p. 6

2.2 HK formulations and normalization

  • HK’s dynamic formulation couples spatial transport and local mass change through a continuity equation with reaction, while its cone formulation gives admissible triples upper bounds on HK².The paper uses the cone or semi-coupling formulation; the diagonal choice is among the admissible constructions described for bounding the distance.Source: §2.2 · p. 7

2.3 Theorem A: existence, mass control, positivity

  • Coercivity yields an exact-step minimizer with bounded mass; if U′(0+) = −∞, its density is positive almost everywhere and has full support.The theorem does not require differentiability of U at zero.Source: §2.3 · p. 8

2.4 Monge representation on domains and tori

  • For positive endpoint densities on the stated domains, an optimal Monge-growth pair exists with positive growth factor and an essentially injective transport map.The result also holds with the source and target measures interchanged.Source: §2.4 · p. 8

2.5 Theorem B: density bounds, the discrete Euler equation, and the metric– dissipation identity

  • The self-consistent chemical potential is nonincreasing along an optimal map, and the new density has two-sided bounds and satisfies the exact discrete Euler–Lagrange equation.The comparison uses essentially injective pairs; with Lipschitz V and W under the stated assumptions, the new density is Lipschitz.Source: §2.5 · pp. 9–10
  • The discrete Euler equation yields HK²(µ1, µ̄) = τ²Ig(ρ1), linking the proximal distance to the metric-dissipation quantity at the new density.The identity follows by substituting the Euler–Lagrange relation into the pointwise HK potential identity.Source: §2.5 · p. 10

3 The Monge-growth formulation

The Monge-growth formulation represents each update by a transport map and a mass-changing factor. Under positive-density hypotheses, unrestricted pair optimization recovers the exact JKO minimum, while cone-action bounds support a one-step descent test.

3.1 Exact reparametrization and pair stability

  • Under positive-density hypotheses for the source and a JKO minimizer, minimizing over admissible pairs attains the exact JKO minimum.An optimal Monge-growth pair attains the pair objective’s infimum.Source: §3.1 · p. 10
  • Every finite exact HK-JKO sequence with positive densities has an essentially injective restarted Monge-growth representation whose cone action equals each step’s squared HK distance.Source: §3.1 · p. 11
  • Pair stability estimates control changes in the induced endpoint measure when the map and growth factor are approximated.With bounded growth factors, pair approximation also yields a one-step state error.Source: §3.1 · p. 11

3.2 Primal action versus restricted duality

  • The cone action upper-bounds squared HK distance, whereas maximizing over a restricted feasible dual class gives a lower bound that need not ensure exact JKO descent.The restricted primal class may overestimate the metric term; the unrestricted optimal pair attains the bound.Source: §3.2 · pp. 11–12
  • The cone action separates into reaction and transport components, whose shares can be monitored alongside energy descent.The identity pair reproduces the source with zero action.Source: §3.2 · p. 11
  • If a pair objective is at least δ below the identity-pair value, its induced state satisfies the one-step descent estimate with slack δ.Summing accepted-step inequalities accumulates verified slacks in the dissipation bound.Source: §3.2 · p. 12
  • The acceptance test is exact only when relevant functional values have controlled quadrature error, and energy decay alone does not rule out a nontrivial pure-reaction update.A restricted dual objective’s descent alone also does not guarantee descent of the exact JKO functional.Source: §3.2 · p. 12

4 Neural realization and algorithm

The neural scheme jointly parameterizes a spatial map and a positive growth factor, with map constructions designed to preserve admissibility. Sampling, mass propagation, and empirical acceptance checks implement the restarted updates, but these checks do not by themselves certify exact descent.

4.1 Pair objective and neural parametrization

  • The map and positive growth factor are jointly parameterized, with the growth network modifying a fixed positive reference profile.The experiments use a unit reference profile and jointly optimize the growth network with the map.Source: §4.1 · p. 13
  • Constrained residual maps use smooth bounded displacements and a contraction condition to ensure orientation-preserving diffeomorphisms of the domain.ReLU is excluded because parameter-dependent derivative jumps can create interface terms missed by pointwise automatic differentiation.Source: §4.1 · pp. 13–14
  • A neural ODE flow is an alternative admissible map construction when its vector field preserves the boundary or is periodic on the torus.The exact flow is orientation preserving, with a controlled positive Jacobian.Source: §4.1 · p. 14

4.2 Sampling, mass propagation, and the algorithm

  • The normalized next-state sampling law must carry weights proportional to the squared growth factor; unweighted mapped samples represent it only when growth is constant.The density at mapped points is updated using the source density, growth factor, and map Jacobian.Source: §4.2 · p. 15
  • The algorithm accepts a candidate only after an empirical objective comparison with the identity pair and required validation checks; without quadrature-error bounds, this does not certify exact descent.Density-model propagation also adds per-step representation error, and its empirical checks do not transfer exact pair-propagation convergence to the trained recursion.Source: §4.2 · pp. 15–17
  • Beyond the cone cutoff, the map gradient vanishes, so training can stall at a pure-reaction update that still passes the descent criterion.Monitoring maximal displacement and transport share, or constraining displacement, can mitigate this degeneracy.Source: §4.2 · p. 17

5 Guarantees for parametric and computed steps

The guarantees distinguish approximation of an exact step from an a posteriori bound for a computed state. Parametric convergence requires approximation of a regular optimal pair, while the computed-state estimate relies on convexity and globally feasible dual information.

5.1 Theorem C: one-step representability

  • Theorem C transfers approximation of a regular optimal Monge-growth pair to convergence of endpoint densities and objective values, with the cone-action overestimate vanishing.The assumption separates regularity of the exact step from the approximation capability of the chosen neural class.Source: §5.1 · p. 18
  • The exact theory does not establish the stronger regularity assumed by Theorem C, and small-time consistency experiments do not establish its fixed-step approximation property.The exact theory establishes essential injectivity; the experiments report decreasing rescaled-field errors and PDE residuals under time-step refinement.Source: §5.1 · p. 18
  • Theorem C extends to any fixed finite number of steps when its regular-pair approximation hypothesis holds at each step.No convergence rate uniform in the number of steps is established.Source: §5.1 · pp. 18–19

5.2 Theorem D: an a posteriori error estimate from a feasible dual pair

  • Theorem D uses a feasible pair’s primal objective and a globally feasible HK dual lower bound to control the computed state’s exact JKO objective gap.It requires exact-step regularity and a positive-semidefinite interaction, but not a parametric approximation hypothesis.Source: §5.2 · p. 19
  • For Boltzmann entropy, the a posteriori estimate also bounds the L1 density error.The theorem additionally bounds an internal-energy convexity remainder.Source: §5.2 · p. 19
  • The computed-state guarantee applies only with positive-semidefinite interactions and global dual feasibility, and numerical use requires rigorous gap evaluation.The estimate controls the induced density, not the map-growth pair or cone action; finite-horizon propagation is not proved.Source: §5.2 · p. 20

6 Numerical experiments

The experiments test one-step PDE consistency, relaxation with both transport and reaction, Gaussian flows, fully implicit interactions, and a finite-grid certificate. Trajectories use fitted positive neural densities and quadrature-based diagnostics; reported comparisons are descriptive rather than statistical significance claims.

6 Numerical experiments

  • Trajectory comparisons use quadrature estimates and common seeds; comparisons are descriptive, and rejected attempts do not advance physical time.Time-dependent error curves omit the first five time levels; ablation endpoints use the latest time attained by all compared trajectories.Source: §6 · p. 20

6.1 One-step PDE consistency and small-time scaling

  • The relative strong PDE residual falls from 0.0176 at τ = 0.02 to 0.000743 at τ = 0.000625; at τ = 0.00125, velocity and reaction field errors are 0.019 and 0.0113.These finite-resolution observations do not establish an asymptotic rate.Source: §6.1 · p. 22, Figure 1 · p. 22, Figure 2 · p. 22

6.2 Entropy–potential relaxation and the role of both pair components

  • At the common ablation time, full-pair, pure-growth, and transport-only trajectory errors are 0.00131, 0.0193, and 3.73, respectively; endpoint energy alone does not rank accuracy.The one- and two-dimensional full-pair errors to finite-volume trajectories are 2.01 × 10−5 at t = 2.5 and 0.000532 at t = 1.2.Source: §6.2 · pp. 23–24, Figure 4 · p. 23, Figure 3 · p. 23

6.3 Gaussian entropy and confinement flows

  • Endpoint relative L2 density errors against whole-space Gaussian references are 0.00803 for eight-dimensional entropy flow at t = 0.15 and 0.00348 for FP2D at t = 1.2.The errors include time-discretization, finite-box, learned-pair, and density-fit errors.Source: §6.3 · p. 25, Figure 6 · p. 25, Figure 7 · p. 26

6.4 Coercive fully implicit interaction benchmark

  • The fully implicit and frozen-field interaction comparison establishes no uniform accuracy advantage for either treatment.The strong-form residual is a grid-dependent finite-difference diagnostic, and the refinement checks do not establish spatial convergence.Source: §6.4 · p. 26, Figure 8 · p. 27

6.5 Finite-grid primal–dual certificate

  • Across tested widths, the largest relative growth-equation RMS and maximum residuals are 9.02×10−5 and 0.000583; observed L1 distances to the numerical grid iterate are at most 0.00853.Measured errors and gaps need not decrease monotonically with width.Source: §6.5 · p. 28, Figure 10 · p. 28, Table 7 · p. 28
  • For the numerical grid reference, the primal upper value and feasible dual lower value are both −0.6267, with gap 2.746×10−9 and L1 radius 0.000118.The certificate is discrete, not a continuum certificate, and feasibility uses a floating-point margin rather than interval arithmetic.Source: §6.5 · p. 28, Table 7 · p. 28

7 Discussion and conclusion

The discussion collects the exact-step theory, conditional recovery under parametric approximation, and a primal–dual error estimate. It also distinguishes these results from convergence guarantees for the trained stochastic solver and from a general τ ↓ 0 theory.

7 Discussion and conclusion

  • For the exact step, the stated results include existence, mass control, positivity, density bounds, a discrete Euler–Lagrange equation, and a metric–dissipation identity on bounded convex domains and flat tori.With Lipschitz V and W, the density is Lipschitz as well.Source: §7 · p. 29
  • One-step recovery, and recovery over each fixed finite horizon by finite induction, require a regular-pair approximation hypothesis and do not imply convergence of the trained stochastic solver.Approximation by the fixed two-block experimental architecture is not proved.Source: §7 · p. 29
  • With positive-semidefinite interactions and a globally feasible dual pair, the primal–dual gap controls objective suboptimality and, for Boltzmann entropy, the L1 density error.The estimate has limitations and needs additional ingredients for finite-horizon propagation; the experiments do not supply continuum quadrature control or step-uniform estimates.Source: §7 · p. 29
  • A general τ ↓ 0 theory additionally requires stepwise estimates uniform in the step index.The experiments test small-step pair structure and a deterministic-grid certificate but do not replace the missing continuum quadrature control or step-uniform estimates.Source: §7 · p. 29

A Proof of Theorem 2.3 (Theorem A)

The proof obtains an exact JKO minimizer by controlling mass, establishing compactness of a minimizing sequence, and passing to the limit. Under an additional entropy condition, it also proves strict positivity almost everywhere.

A Proof of Theorem 2.3 (Theorem A)

  • Superlinearity of the internal energy yields uniform integrability of minimizing-sequence densities, and weak L1 compactness gives a density-valued limit without a singular part.The proof then uses lower semicontinuity of the energy and HK distance to pass to the limit.Source: A Proof of Theorem 2.3 (Theorem A) · pp. 30–31
  • Coercivity of the scalar mass bound gives a finite upper bound on the mass of every minimizer.The scalar sublevel contains zero and is bounded by coercivity.Source: A Proof of Theorem 2.3 (Theorem A) · p. 31
  • If U′(0+) = −∞, a minimizer cannot vanish on a set of positive Lebesgue measure, so its density is positive almost everywhere and its support is all of Ω.The proof perturbs the density on a proposed zero set and contradicts minimality.Source: A Proof of Theorem 2.3 (Theorem A) · pp. 31–32

B Proof of Proposition 2.5

The proof establishes a Monge-growth representation and essential injectivity for positive-density measures in both Euclidean domains and flat tori. The torus argument obtains injectivity by comparing forward and reversed optimizers.

B Proof of Proposition 2.5

  • In the Euclidean case, positive densities and convexity allow the cited Monge theorem to yield a Monge-growth representation, positive growth factor, and essential injectivity.The measures are strongly reduced, so no non-reduced marginal part must be discarded.Source: B Proof of Proposition 2.5 · p. 32
  • On the flat torus, entropy–transport theory gives finite Lipschitz logarithmic potentials and a unique optimizer concentrated on the graph of a Riemannian c-exponential map.The corresponding growth factor gives the Monge formulation identities.Source: B Proof of Proposition 2.5 · p. 33
  • Comparing the forward optimizer with the reversed optimizer shows that the torus map is essentially injective with respect to the source measure.Uniqueness makes the transposed optimizer a reverse graph, yielding an inverse relation almost everywhere.Source: B Proof of Proposition 2.5 · p. 33

C Proof of Theorem 2.6 (Theorem B)

The appendix fixes notation and warns that potential symbols and transport-pair orientations differ across its subsections.

C Proof of Theorem 2.6 (Theorem B)

C.1 Monotonicity of the discrete chemical potential along the optimal map and density bounds

The section establishes monotonicity of the discrete chemical potential along an optimal map and derives upper and lower density bounds. Its comparison argument uses the self-consistent chemical potential, with field oscillation entering the density estimates.

C.1 Monotonicity of the discrete chemical potential along the optimal map and density bounds

  • The comparison scheme adapts [13, Sec. 4], replacing U′(ρ1) with the self-consistent chemical potential and introducing a field-oscillation loss in the density bounds.Lemma C.4 and Propositions C.5–C.6 correspond to [13, Lem. 4.1 and Props. 4.2–4.3].Source: §C.1 · p. 34
  • The discrete chemical potential is monotone along the optimal map under the subsection’s assumptions.The comparison argument localizes any strict violation and constructs a target-replacement competitor that does not increase the HK cost.Source: §C.1 · pp. 34–35
  • The upper density bound follows by excluding high-chemical-potential sets, with the bound depending on the energy derivative and self-consistent field.The proof uses backward invariance of a high-potential set under the optimal map and a growth-factor competitor.Source: §C.1 · pp. 36–37
  • A lower density bound is obtained by excluding low-potential sets, using a split competitor that adds mass through a diagonal branch.The stated lower bound uses U′(0+) = −∞ and includes the field-oscillation loss.Source: §C.1 · pp. 37–39

C.2 Dual potential and discrete Euler equation

This section establishes regularity and identities for HK dual potentials and Monge pairs, deriving the fully implicit step’s discrete Euler–Lagrange equation and metric-dissipation identity. It also shows that neural approximations of a regular optimal Monge-growth pair recover its induced density uniformly and approach the exact JKO objective, with primal majorization transferring convergence to the JKO gap and cone action.

C.2 Dual potential and discrete Euler equation

  • A saturated optimal transformed dual pair has W^{1,∞} components, a source component unique µ-almost everywhere, and an associated Monge map uniformly below the π/2 cutoff.The bounds can be chosen uniformly over density pairs with the same two-sided bounds m and M.Source: §C.2 · pp. 40–41
  • For an optimal Monge-growth pair, the saturated potentials satisfy 1 − ψ(x) = q(x) cos r(x) and 1 − χ(T(x)) = cos r(x)/q(x), with r the transport distance.The source potential also obeys ∇ψ = −2q sin(r)e, where e points along the minimizing geodesic from x to T(x).Source: §C.2 · pp. 41, 43
  • Under the stated assumptions, the fully implicit HK-JKO minimizer satisfies a discrete Euler–Lagrange identity, and its full chemical potential has a globally Lipschitz representative.Applying the identity to the reversed pair (µ1, µ0) yields a Lipschitz representative of Φ1 = −ϕ1/τ; additional assumptions on V and W imply ρ1 ∈ W^{1,∞}(Ω).Source: §C.2 · pp. 43–45
  • The discrete metric–dissipation identity, combined with one-step minimality, gives the stated energy-dissipation inequality for the HK-JKO step.The comparison uses 2τ HK²(µ1, ¯µ) = Jτ(µ1 | ¯µ) ≤ F(¯µ).Source: §C.2 · pp. 44–45

D Proof of Theorem 5.2 and Corollary D.2 (Theorem C)

  • If continuous bijections S_n on a compact space converge uniformly to a continuous bijection S, their inverse maps converge uniformly to S^{-1}.The proof argues by contradiction, using compactness and injectivity of S.Source: D Proof of Theorem 5.2 and Corollary… · p. 45
  • If bijections converge uniformly to a map with continuous inverse on the compact domain, their inverses converge uniformly as well.The proof uses compactness and continuity to rule out a sequence of points where inverse-map convergence fails.Source: D Proof of Theorem 5.2 and Corollary… · p. 45

D.1 One step

  • Neural maps converge to the regular optimal map in C1 and growth factors converge uniformly, yielding uniform recovery of the induced density.The argument uses positivity of the optimal map’s Jacobian to ensure the approximating maps also have Jacobians bounded away from zero for sufficiently large n.Source: §D.1 · pp. 45–46
  • Uniform convergence of maps, amplitudes, densities, and Jacobians makes the pair objective converge to the exact JKO minimum, with a nonnegative gap tending to zero.Continuity of the energy terms on the relevant compact sets and uniform convergence of the integrand support passage to the objective limit.Source: §D.1 · p. 46
  • The primal upper bound bounds the original JKO gap by the pair-objective gap and controls excess cone action, so sufficiently large n proves (5.2)–(5.5).Source: §D.1 · p. 46

D.2 A fixed finite horizon

Under the regularity and network recovery assumptions, neural pairs can approximate exact HK-JKO iterates over a fixed finite horizon. The represented states, variational objectives, and cone-action components converge uniformly across the steps.

D.2 A fixed finite horizon

  • Under Assumption 5.1 for each selected optimal pair, neural pairs and recursively represented states converge over the fixed finite horizon.The finite-step construction synchronizes the one-step approximations and propagates state error recursively.Source: §D.2 · pp. 46–47
  • Both variational objectives are recovered uniformly over the horizon, and the cone action converges to the exact-step action.The original JKO objectives converge through density convergence and the HK triangle inequality.Source: §D.2 · p. 47
  • The transport and reaction components of the cone action also converge uniformly over the horizon.Source: §D.2 · pp. 47–48

E Proof of Theorem 5.4 and Corollary 5.5 (Theorem D)

The appendix develops a primal–dual certificate for JKO-step accuracy and details conditions and limitations for potential-generated pair approximations and step-uniform estimates.

E Proof of Theorem 5.4 and Corollary 5.5 (Theorem D)

  • The primal–dual gap bounds objective suboptimality and, for Boltzmann entropy, controls the L1 density error under the theorem’s assumptions.The proof constructs a dual lower bound using global dual feasibility and positive-semidefinite interaction, then compares it with the primal cone-action value.Source: E Proof of Theorem 5.4 and Corollary… · pp. 48–49
  • Positive-semidefinite interactions support the tangent lower bound used to construct a pointwise-separable dual certificate, while strict convexity of U gives a unique exact-step minimizer.The certificate minimizes the lower bound through the convex conjugate of U.Source: E Proof of Theorem 5.4 and Corollary… · p. 48
  • The gap decomposition separates a nonnegative discrete Euler residual from the primal–dual metric gap.The two components are nonnegative by Fenchel–Young and primal–dual metric bounds.Source: E Proof of Theorem 5.4 and Corollary… · pp. 49–50

F.1 Potential-generated pairs and admissibility

  • The forward predictor is only a first-order calculation, not a consistency estimate for the exact minimizer, and the C2 regularity needed for approximation remains open.The exact-step theory supplies only Lipschitz regularity; the Euler–Lagrange equation gives no elliptic regularity gain.Source: §F.1 · pp. 50–51
  • Potential-generated pairs match the structure of exact optimal pairs but require domain, injectivity, and minimizing-branch conditions; arbitrary network outputs are not guaranteed admissible.Approximation also requires a regular C2 potential and a network sequence converging in C2 while preserving admissibility.Source: §F.1 · pp. 50–51

F.2 Step scaling and uniform estimates

  • Direct iteration of the Boltzmann-entropy density bound incurs a factor e^ω1 per step and yields no τ-independent bound over Nτ steps without additional τ-scaled control.The main-text results do not require this scaling or step-uniform bounds.Source: §F.2 · p. 51

F.3 Experimental implementation and refinement checks

The experiments use neural maps, growth factors, and numerical checks to assess pair optimization and propagated trajectories. The reported interaction residuals are discretization-sensitive and do not establish a grid-converged strong-form residual.

F.3 Experimental implementation and refinement checks

  • Trajectory pairs are initialized by fitting map and growth outputs to the source predictor before joint training, while independent pair-scaling and certificate tests start from (id, 1).Recursive propagation fits the log density, restores sampled mass, and uses the raw inverse-map pushforward for certification.Source: §F.3 · pp. 51–52
  • The entropy objective is estimated with an integration-by-parts identity using the current source score; this changes finite-quadrature values and gradients but not the continuum objective.Trainable networks use float32, while quadrature and linear solves use CPU float64.Source: §F.3 · p. 52
  • All six stored trajectories satisfy the reported per-step energy-descent check, but these checks do not bound accumulated projection error.Table 10 reports fit and mass errors as maxima over steps and work as mean closure evaluations, not wall-clock time.Source: §F.3 · p. 53, Table 10 · p. 53
  • At T = 0.3 on the 128^2 grid, the implicit-state residual numerator is about 0.14% of either individual transport-term norm, but the nearest-image kernel has derivative kinks.Diffusion and drift each have L2 norm about 2.6, while their sum has norm about 0.03; smooth-kernel conclusions are not invoked.Source: §F.3 · p. 53
  • The 64^2-grid implicit-state residual is 0.048, and residuals first decrease then increase under refinement, so no grid-converged strong-form residual is established.The coarse-grid reference differs from the 128^2 reference at quarter time step by 6.957×10^-4 in relative L2.Source: §F.3 · p. 53

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