2026-09-02
An advection PDE from conserved denoiser identity recovers Flow Matching's straight lines. A 1D two-mode toy peaks at |df/dt| 9.23 near crossings vs 2.65 far away.
The usual derivation of Flow Matching and Rectified Flow is Eulerian. You stand at a fixed coordinate, watch probability mass being carried by a velocity field, and let the continuity equation keep that mass conserved. Optimal transport then supplies the straight couplings. That view guarantees the generated distribution. It says almost nothing about what a discrete ODE solver actually hits at each step.
The solver sees a denoiser output. In ordinary diffusion, a nonlinear noise schedule warps the intermediate marginals, so the predicted clean image moves at every step. A finite step splits into predict and correct: evaluate f(x,t), step by the residual x−f, then ask the denoiser again at (x+Δx, t−Δt). A first-order Taylor expansion says the new prediction is the old one plus Jf Δx minus a temporal-drift term. The spatial Jacobian patches local geometry; the time derivative chases a receding clock. A moving target forbids large steps. Thousands of diffusion steps are the default.
The harder question is which trajectories keep the target still enough for large steps, and why empirical Flow Matching still needs about 20 to 50 steps plus distillation to reach one-step generation.
On an ideal path, the denoiser f(x,t) names the clean sample x0 at the first noise level and never changes its mind. The total time derivative along the path is zero. That is a Lagrangian view: sit inside one particle and ride. Geometrically the ideal denoiser is the inverse flow map, f(x,t)=x0, so the particle's target identity is conserved.
The chain rule turns the condition into a first-order quasilinear advection PDE: any temporal drift ∂f/∂t must be cancelled by the spatial Jacobian Jf acting on the velocity v. The PDE constrains the pair (f,v) and does not pick a unique velocity. The paper therefore posits a residual field, motivated by Tweedie's formula and residual flow: v(x,t)=c(t)(x−f(x,t)), with c(t) an arbitrary scalar schedule. Velocity is driven entirely by how far the current state sits from the denoiser prediction.
The method of characteristics tests self-consistency. It does not derive that ansatz. Imposing df/dt=0 along the curve forces f to be constant; the boundary f(x,0)=x0 pins the constant. Substituting back yields a separable linear ODE whose solution is X(t)=x0+σ(t)z1, with σ(t)=exp(∫c(t)dt) and z1 the noise at t=1. Those are the straight interpolants of Flow Matching.
The claim is deliberately narrow. The residual field is not shown to be the unique solution of the PDE; other velocities could in principle be made consistent. Three things are shown: the residual ansatz can be reconciled with invariance; reconciliation forces f constant along characteristics; the t=0 boundary then pins the trajectories to straight lines.
Curvature has a single mechanical source. Off the characteristic, ∂f/∂t=−Jf v. For an optimal MMSE denoiser, or an ideal denoiser with symmetric Jacobian, Jf is proportional to the posterior covariance Σpost. Covariance is positive semidefinite, so v⊤∂f/∂t=−v⊤Jf v≤0. The target recedes along the direction of travel. Curved solvers have to take tiny steps. As t→0 the data manifold sharpens and the eigenvalues of Jf diverge; a curved trajectory hits that explosion. On straight characteristics ∂f/∂t=0 and the Jacobian penalty is cancelled.
Training tears this picture. Independent straight pairings of noise and data cross in high dimension. A deterministic network cannot output two targets at one point, so it averages. Posterior covariance explodes at the crossing, trajectories bend, and baseline Flow Matching still needs about 20 to 50 solver steps. Reflow distillation rolls the baseline to produce non-crossing pairs, then trains a new model on those pairs. Conflicts gone, Σpost→0, Jf→0, the drift mechanism is starved, and one-step generation becomes possible.
A 1D two-mode toy pins the numbers. Data is x0∈{−1,+1}, linearly interpolated with standard Gaussian noise. Each family of characteristics is straight, but the shared noise makes them overlap densely near x=0 when t is close to 1. The Bayes-optimal denoiser is closed form: f(x,t)=tanh((1−t)x/t²), v=(x−f)/t. At t=0.9, f is nearly linear, averaging hard. At t=0.1 it collapses to a step at x=0, so the Jacobian diverges on the crossing coordinate.
| start | peak df/dt | path |
| z1=±2.0, far from the crossing | ≈2.65 | nearly straight |
| z1=±0.05, near the crossing | ≈9.23 | S-curve, lingering near x≈0 |
| z1=0, decision boundary | never resolves | f frozen at 0 |
Peak drift near the crossing is about 3.5 times the far-start value, and the peak sits around t≈0.2 to 0.5. The z1=0 case is the toy version of a low-step sample that blurs between modes.
Eulerian objects vt and pt cannot be read out at sample time. The network emits the denoiser tensor f. Building the invariant as df/dt=0 gives inference a monitor: if consecutive outputs swing wildly, the advection PDE is being violated and that sample is still fighting unresolved crossings. Slow the solver there, or aim distillation at that region of the manifold. A blurry one-step image, in this picture, means characteristics still cross and averaging has wiped the target.
This is not a new training objective. It recasts known facts into a chain a solver can check: why straight flows take large steps, why diffusion cannot, why Reflow can go to one step. For teams already running Flow Matching, the value is a diagnostic lens. Consistency models already used a similar invariance as a finite-difference training loss; this paper writes it as a full PDE and isolates the Jacobian as the curvature source.
The velocity is not derived from the PDE. It is inserted as an ansatz and then checked for consistency. Straight lines are not the unique corollary of the invariance principle.
The experiment stops at a 1D two-point mixture with a closed-form posterior. Real image manifolds have continuously many modes, so the diagnostic has to be read as a distribution over samples. The paper never reports that quantity on a real image model. The 20-to-50-step and thousand-step figures are literature rules of thumb, not measurements in this work. Jf∝Σpost assumes an ideal MMSE denoiser with symmetric Jacobian; a trained network need not obey that. How often paths physically cross in high dimension is left unquantified.