Beckmann Transport Models: From Autonomous Flows to One-Step Maps

Affiliations: Harvard University, Capital Fund Management, University of Oxford, New York University, University of Pennsylvania

Publication: arXiv:2608.01692v3 [cs.LG], 13 Aug 2026

The paper proposes "an instantiation of flow matching that relies on a time-independent velocity field—an autonomous flow—to exactly map between two distributions when the target is supported on a sufficiently lower-dimensional data manifold. The authors also show that the associated one-step generative map is constant in the direction of the autonomous velocity field and therefore satisfies a simple conservation equation, which motivates learning the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem, and the construction extends beyond flow matching to other probability fluxes satisfying the same constraint. The framework includes a closed-form Coulomb transport closely related to Poisson Flow and a flow-matching-consistent version of Equilibrium Matching. The authors validate their approach on ImageNet 256 × 256."

Standard flow matching and diffusion models drive generation by transporting samples from a base distribution µ0 to a target distribution µ1 using a time-dependent velocity field (or drift) learned by quadratic regression. The paper explores an intriguing alternative: constraining the drift to be time-independent, minimizing the same regression loss over fields b: Rd → Rd that do not depend on t, and performing generation using the autonomous equation Ẋt = b(Xt) solved with initial X0 ∼ µ0. This is essentially the construction proposed by Wang and Du [2025] under the name Equilibrium Matching (EqM).

The authors note that the original EqM paper does not establish this transport property, and its proposed loss need not enforce the required source–sink balance. The paper provides a rigorous justification under explicit geometric and regularity conditions. Specifically, when µ1 is supported on a manifold M1 ⊂ Rd of codimension at least two, the time-independent flow matching drift defines a valid transport: trajectories of the autonomous flow converge to M1, and the resulting map pushes µ0 forward to µ1.

The proof uses the finite FM current and the divergence equation satisfied by the drift. The endpoint map T is selected by following each autonomous trajectory to its endpoint and then freezing it there; it satisfies a simple conservation equation along its characteristics. This equation motivates a stop-gradient objective for training T directly from samples.

The paper's main contributions are:

1. Proof of valid transport for time-independent FM drift: "Under the regularity and codimension conditions of Assumptions 1–2, the time-independent FM drift defines a valid transport: the minimizer of the FM regression loss over time-independent fields b generates a flow Ẋt = b(Xt) that pushes µ0 to µ1 (Proposition 1). This gives a flow-matching-consistent correction of Equilibrium Matching."

2. Weighted Beckmann current transport: "For a finite positive weight ν, we show that the unique minimizer of a weighted Beckmann current problem defines a valid autonomous transport when it points uniformly toward the target near M1 (Theorem 1). Gradient-constrained FM provides a natural route to this minimizing current; the canonical Coulomb field with ν ≡ 1 is treated separately (Proposition 2)."

3. Eulerian equation for the flow map: The map x ↦ Xt(x), which gives the position at time t of the autonomous trajectory starting from x, satisfies the Eulerian equation ∂tXt = b · ∇Xt (Proposition 3).

4. Conservation equation for the one-step map: "The endpoint of each trajectory defines the one-step map T(x), which pushes µ0 forward to µ1 (Proposition 1). This map is constant in the direction of the autonomous field b, so that b · ∇T = 0, with T = id on M1 (Theorem 2)."

5. Stop-gradient objective for direct map learning: "The conservation equation b · ∇T = 0 motivates a stop-gradient objective for direct map learning. We prove that the population stop-gradient update vanishes at the true map T (Theorem 3) and evaluate the resulting one-step and iterated inference procedures experimentally."

The paper situates itself relative to several research threads:

Flow matching, diffusion, and equilibrium matching: Flow matching and diffusion models construct differential equations with time-dependent drifts that transport between base and target distributions. The BTM drift minimizes the same FM loss, but over time-independent fields. BTM establishes an exact transport result for a flow-matching-consistent version of Equilibrium Matching and shows how the original loss can produce a biased endpoint distribution. Relatedly, Christensen et al. construct time-homogeneous denoising diffusions using Doob h-transforms with random horizons, but their construction is stochastic and score-based, rather than a deterministic transport derived from a Beckmann current.

Poisson Flow: The choice ν = 1 also yields a closed-form Coulomb transport related to Poisson Flow. In the BTM construction, the field is generated by the signed source–sink distribution µ0 − µ1, whereas PFGM places the data charges on an augmented hyperplane and uses the induced source distribution on a distant hemisphere.

Noise-unconditioned models: Kadkhodaie and Simoncelli provide a blind-denoiser sampler, while Sun et al. study removing noise conditioning. For linear FM, their population target matches our autonomous drift up to orientation, but their analysis bounds error relative to a conditioned sampler rather than proving an exact pushforward.

Optimal transport: The divergence condition is classical in OT. Standard flow matching realizes a transport without minimizing the Benamou–Brenier action; analogously, the FM instance of BTM gives a dynamical realization of a Beckmann flux without minimizing the associated cost.

Consistency models and one-step generation: One-step generation has been pursued through consistency models, rectified flow, Flow Map Matching, MeanFlow, and Shortcut Models. These methods start from an underlying time-dependent flow and distill or self-distill it into a one-step network. Drifting learns a one-step map directly, by evolving the network's pushforward during training, using an MMD distance to measure progress. BTM also learns the map directly, but exploits a different structural fact: the terminal map of the autonomous flow satisfies a stationary conservation equation along characteristics.

Standard flow matching constructs a stochastic interpolant:

Is = αsx0 + βsx1, x0 ∼ µ0, x1 ∼ µ1, s ∈ [0, 1],

where αs and βs satisfy boundary conditions α0 = β1 = 1, α1 = β0 = 0. The probability distribution µs of Is coincides with that of the solution to the probability flow ODE Ẏs = bs(Ys), Y0 ∼ µ0, provided the time-dependent drift is obtained by minimizing the standard FM loss over functions b̂s(x) that depend on s.

The paper considers minimizing the same averaged FM loss over time-independent functions b̂(x):

b = arg min Es∼U[0,1],x0,x1 b̂(Is) − İs2,

and considers the autonomous ODE Ẋt(x0) = b(Xt(x0)), X0(x0) = x0. The population minimizer is b(x) = Ex0,x1,s[İs Is = x].

Assumption 1 (Regular source and singular target, d ≥ 2): "The base distribution µ0(dx) = ρ0(x) dx has a smooth, strictly positive, rapidly decaying density; in particular, ρ0 has finite moments of every order. The target distribution µ1 is supported on a compact smooth embedded k-dimensional manifold M1 ⊂ Rd without boundary, where k ≤ d − 2, and has a smooth, strictly positive density ρ1 on M1 with respect to its natural volume measure vol M1."

Assumption 2 (Straight FM interpolant with controlled endpoint behavior): "The interpolant has straight-line geometry, Is = αsx0 + (1 − αs)x1, where x0 ∼ µ0 and x1 ∼ µ1 are independent, and α ∈ C1([0, 1]), α0 = 1, α1 = 0, and α̇s < 0 for 0 ≤ s 0 and γ ≥ 0."

"Under Assumptions 1–2, for µ0-almost every x0 ∈ Rd, there exists a finite time τ(x0) > 0 such that the solution of (6) is defined and remains in Rd M1 for 0 ≤ t < τ(x0), and T(x0):= lim Xt(x0) as t↑τ(x0) exists and belongs to M1. In addition, the resulting endpoint map T satisfies Tµ0 = µ1."

Proof sketch: Integrating the continuity equation from s = 0 to s = 1 gives ∇ · j = µ0 − µ1, where j = ∫01 bsµs ds is the time-averaged probability flux. The minimizer b satisfies j = νb, where ν = ∫01 µs ds > 0. Hence ∇ · (νb) = µ0 − µ1. Since ν > 0, the fields j and b have the same oriented flow lines. Away from M1, ∇ · j = ρ0 > 0, so volumes transported by j expand. The FM current carries no mass to infinity, and near M1 it points toward the target. Hence almost every trajectory ends on M1. For A ⊆ M1, flux balance equates source mass inside the basin with sink mass at its endpoint: µ0(B A) = µ1(A), giving Tµ0 = µ1.

"The role of the singular target is visible from the divergence equation (7): it provides the localized sink at which the autonomous trajectories can end. Codimension also matters. In codimension one, a current satisfying the same divergence equation may pass through the target rather than stop there, and counterexamples show that the source–sink balance alone is not sufficient."

"There are two distinct clocks. The schedule αs, with s ∈ [0, 1], governs the interpolant Is and the original time-dependent flow Ẏs = bs(Ys). The autonomous flow (6) instead runs on its own clock t. Its oriented streamlines, and hence the transport map T, depend only on the straight-line geometry ax0 + (1 − a)x1, not on the interpolant clock."

The behavior of αs near s = 1 determines the magnitude of the autonomous drift near the target:

• For α̇1 0): b(x) ≍ r(x)(γ/(γ+1)) → 0 near M1, with finite hitting time.

For a fixed positive weight ν of finite mass, consider the affine class:

J ν:= j ∈ L2(ν−1 dx; Rd): ∇ · j = µ0 − µ1,

where the divergence equality is understood distributionally. The weighted Beckmann problem:

j ν:= arg min (1/2)∫ Rd j(x)2/ν(x) dx

has a unique solution whenever J ν is nonempty. Set b:= j ν/ν.

"Under Assumption 1, let ν ∈ C1(Rd M1) be strictly positive and satisfy 0 < ∫ Rd ν(x) dx < ∞. Assume that J ν is nonempty, let j ν be the minimizer in (9), and suppose that b:= j ν/ν belongs to C1(Rd M1). Suppose moreover that there exist r0, κ > 0 such that j ν(x) · ∇r(x) ≤ −κj ν(x) < 0 for 0 < r(x) < r0. Then, for µ0-almost every x0, there exists τ(x0) ∈ (0, ∞] such that the solution of (6) is defined and remains in Rd M1 for 0 ≤ t < τ(x0), and the unique limit T(x0):= lim Xt(x0) as t↑τ(x0) belongs to M1, where t ↑ τ means t → ∞ when τ = ∞. In addition, the resulting endpoint map T satisfies Tµ0 = µ1."

Let j FM be the FM current of Proposition 1 and let ν = ∫01 µs ds be its occupation weight. Then ∫ Rd ν dx = 1, j FM ∈ J ν, and j FM has finite weighted energy. If the autonomous FM regression is restricted to gradient fields b = ∇φ, its population loss is, up to an additive constant, (1/2)∫ Rd ν∇φ2 dx − ∫ Rd j FM · ∇φ dx + (1/2)E[İs2]. If this restricted problem admits a sufficiently regular minimizer, its first-order condition gives ∇ · (ν∇φ) = µ0 − µ1. Since ν∇φ is then feasible and is orthogonal in L2(ν−1 dx) to every finite-energy divergence-free perturbation, it coincides with the unique minimizer j ν.

"Under Assumption 1, let ν ≡ 1 and let b = ∇φ be the canonical Coulomb field satisfying ∆φ = µ0 − µ1, equivalently b(x) = (1/ω d)∫ Rd (x − y)/x − yd (µ0 − µ1)(dy), for x ∉ M1, where ω d is the surface area of the unit sphere in Rd. Then the first-hitting map of the autonomous flow Ẋt = b(Xt) transports µ0 to µ1, and its hitting time is finite for µ0-almost every initial point."

The finite-mass theorem does not include the Coulomb choice ν ≡ 1 because both relevant finiteness conditions fail: ∫ Rd ν dx = ∞, and the behavior j(x) ≍ r(x)(1−(d−k)) near M1 makes ∫ Rd j2 dx diverge. Nevertheless, it obeys the same transport conclusion because its neutral far field gives no loss at infinity and its singular part points uniformly toward M1.

The classical Beckmann transportation problem is:

B(µ0, µ1):= inf∇·j=µ0−µ1 ∫ Rd j(x) dx = W1(µ0, µ1).

The finite-mass currents constructed in this paper satisfy the same divergence constraint and are therefore competitors in (14), but we do not claim that they solve this L1 minimization problem.

Problem (9) minimizes a quadratic functional with the weight ν fixed. For every fixed current j, inf ν≥0, ∫ν dx=1 ∫ j2/ν dx = (∫ j dx)2, with equality for ν ∝ j. Thus optimizing both the weight and the current would recover the square of the classical Beckmann problem.

The structural analogy is:

• Monge–Kantorovich optimal transport ↔ Benamou–Brenier (time-dependent flow)

• Beckmann transportation ↔ Beckmann Transport Models (autonomous flow)

For the unconstrained FM current j FM, with b FM = j FM/ν and ∫ Rd ν dx = 1:

W12(µ0, µ1) ≤ (∫ Rd j FM dx)2 ≤ ∫ Rd j FM2/ν dx = ∫ Rd b FM2ν dx ≤ ∫01∫ Rd bs2 dµs ds.

Thus the FM current provides an upper bound on the value of the classical Beckmann problem, rather than solving it.

When τ(x0) < ∞, the flow is extended to all later times by freezing after arrival: Xt(x0):= T(x0) for t ≥ τ(x0). Under this convention, Xt inherits the semigroup property X t+s = Xt ∘ Xs for all s, t ≥ 0, with X0 = id.

"Let b be one of the autonomous drifts in Proposition 1, Theorem 1, or Proposition 2, and let Xt be the corresponding autonomous flow (6), with the freezing convention above. Viewed as a flow map of the initial condition x ∈ Rd, Xt(x) satisfies, wherever the spatial derivative exists and for almost every t, the Eulerian equation ∂tXt(x) = b(x)·∇Xt(x), X0(x) = x, for x ∉ M1, Xt(x) = x for x ∈ M1."

Proof: The semigroup identity gives X t+s(x) = Xt(Xs(x)). Differentiating in s at s = 0 yields ∂tXt(x) = b(x) · ∇Xt(x). For t τ(x), the left-hand side vanishes by freezing and the same semigroup argument gives b · ∇Xt = 0.

"Let T be the endpoint map of Proposition 1, Theorem 1, or Proposition 2. The endpoint is constant along µ0-almost every characteristic: T(Xt(x)) = T(x), 0 ≤ t < τ(x). After setting T(x) = x on M1, this gives, wherever T is differentiable, b(x) · ∇T(x) = 0 for x ∉ M1, T(x) = x for x ∈ M1."

The terminal-limit construction uniquely defines T up to a µ0-null set. "By…

Based on the paper, here are specific improvements I can implement in AI systems, along with the resulting capabilities:

Improvement: Replace time-conditioned diffusion/flow-matching models with time-independent velocity fields that satisfy the divergence equation ∇·(νb) = μ0 − μ1. This eliminates the need for time embeddings in neural networks and simplifies the ODE solver.

Capability: A generative model that:

• Uses a single static vector field instead of a time-dependent one

• Requires no time-conditioning in the network architecture

• Produces samples by integrating a simple autonomous ODE until hitting the target manifold

• Achieves exact transport to the target distribution when the target is supported on a codimension-≥2 manifold

These improvements collectively enable: (1) simpler architectures without time conditioning, (2) one-step generation with optional iterative refinement, (3) exact mass preservation for singular targets, (4) training-free baselines for validation, and (5) a principled framework for discrete output generation.

We propose an instantiation of flow matching that relies on a time-independent velocity field (an autonomous flow) to exactly map between two distributions, so long as the target is singular, i.e. supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.