Benamou-Brenier formula

The Benamou-Brenier formula (2000) recasts the Monge-Kantorovich optimal transport problem as a fluid dynamics control problem. Instead of seeking a static coupling between two distributions, it finds the velocity field v(x,t) that transports density ρ₀ to ρ₁ with minimal kinetic energy:

${\inf }_{(\rho ,v)}{\int }_{{\mathbb{R}}^n}{\int }_0^1\frac{1}{2}\Vert v(x,t){\Vert }^2\rho (t,x)dtdx$

subject to the continuity equation ∂ρ/∂t + ∇·(vρ) = 0, with boundary conditions ρ(0,x) = ρ₀(x) and ρ(1,y) = ρ₁(y).

The optimal velocity field is v*(x,t) = ∇ψ(x,t) where ψ satisfies the Hamilton-Jacobi equation ∂ψ/∂t + ½||∇ψ||² = 0. The resulting flow follows straight-line geodesics in Wasserstein space.

This formulation has deep connections to the Schrödinger bridge problem. As shown by Chen, Georgiou & Pavon (2014), the fluid dynamic version of the SB problem has the identical structure but with an additional Fisher information term: the SB minimizes ∫∫[½||v||² + ⅛||∇ln ρ||²]ρ dtdx under the same constraints. The two problems thus differ by a Fisher information penalty, revealing their relationship without zero-noise limits.

In generative modelling, rectified flow can be seen as a learned approximation to the Benamou-Brenier velocity field, and the probability flow ODE provides the deterministic transport path.

Key Details

• Reformulates static OT as a dynamic fluid control problem
• Optimal velocity is a gradient flow: v* = ∇ψ
• ψ satisfies the Hamilton-Jacobi equation
• The Wasserstein-2 distance equals the minimum kinetic energy
• Differs from the SB fluid dynamic formulation by a Fisher information term
• Schrödinger analogue (Léonard 2014, Prop. 4.1): inf $(S)=H({\mu }_0|m)+\int |\nabla {\psi }_t{|}^2/2{\mu }_t(dx)dt$, where $\psi$ is the Schrödinger potential. This Γ-converges to the Benamou-Brenier kinetic action
• Discrete analogue (Léonard 2014, Prop. 4.2): on graphs, $\int {\theta }^{\ast }(j^{\psi }-1){\nu }_t(dx)J_x(dy)dt$ with ${\theta }^{\ast }(b)=(b+1)\log (b+1)-b$
• The displacement interpolation $[{\mu }_0,{\mu }_1]$ is $\mu =\nu$ at the optimum
• Originally from Benamou & Brenier, “A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem” (2000)

Textbook References

Optimal Transport Old and New (Villani, 2009)

• Example 7.35 (p. 157): For quadratic Hamiltonian $H=|p{|}^2/2$, the forward Hamilton-Jacobi semigroup $S_{+}(t,x)=H_{+}^{0,t}{\psi }_0$ solves ${\partial }_{t}S+|\nabla S{|}^2/2=0$ (viscosity sense). The optimal velocity field is $v=\nabla S$, so characteristics are straight lines
• Eqs. (7.33)-(7.34) (p. 171): The action on $P(\mathcal{X})$ is $\mathcal{A}(\mu )={\inf }_v{\int }_0^1\int |v{|}^{2}d{\mu }_{t}dt$ subject to ${\partial }_t\mu +\nabla \cdot (v\mu )=0$, and $W_2^2=\inf \mathcal{A}(\mu )$
• Bibliographical note (p. 172): Nelson’s stochastic mechanics as precursor — action minimisation $\inf \mathbb{E}{\int }_0^1|{\dot{X}}_t{|}^{2}dt$ with $d{X}_t=\sigma d{B}_t+\xi (t,X_t)dt$. This is the stochastic analogue directly relevant to diffusion bridges

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