Dynamic Benamou–Brenier Formulations

• Dynamic Benamou–Brenier formulations are PDE-constrained variational principles that define geodesic distances between measures via action-minimization under continuity equations.
• They extend classical optimal transport by incorporating mass-preserving (or mass-varying) flows and accommodating reactions, multi-species effects, and geometric constraints.
• The framework underpins efficient numerical algorithms such as proximal splitting, ADMM, and interior-point methods for solving high-dimensional optimal transport problems.

Dynamic Benamou–Brenier formulations provide a PDE-constrained variational principle for optimal transport and related problems, characterizing the geodesic distance between measures via a minimization over time-dependent mass and momentum evolutions. Originating in fluid mechanics, this framework generalizes the static Monge–Kantorovich problem by replacing transport plans with mass-preserving (or mass-varying) flows, optimized for least-action subject to the continuity equation. Dynamic Benamou–Brenier formulations admit extensions incorporating reaction, multi-species, entropic, stochastic, and geometric effects, and are central in computational algorithms, mean-field game theory, and the analysis of gradient flows on measure spaces.

1. Classical Benamou–Brenier Formulation: Core Structure and Extensions

The classical setting on a bounded convex $\Omega \subset \mathbb{R}^d$ with prescribed initial $\rho_0$, final $\rho_1$ density (equal mass), introduces the quadratic-cost dynamic optimal transport problem in the Benamou-Brenier form: $$\min_{\rho \ge 0,\, v} \int_0^1 \int_\Omega \frac{1}{2}\|v(t, x)\|^2\, \rho(t, x)\, dx\, dt$$ subject to the continuity (mass-conservation) equation,

$$\partial_t \rho + \nabla_x \cdot (\rho v) = 0, \quad \rho(0, x) = \rho_0(x),\ \rho(1, x) = \rho_1(x), \ v \cdot n|_{\partial \Omega} = 0.$$

Introducing the momentum $m = \rho v$ yields a convex reformulation,

$$\min_{\rho > 0,\, m} \int_0^1 \int_\Omega \frac{\|m\|^2}{2 \rho}\, dx\, dt\quad \text{s.t.}\quad\partial_t \rho + \nabla \cdot m = 0.$$

This dynamic variational problem defines the $L^2$-Wasserstein squared distance and is the archetype for generalized dynamic optimal transport. The action-minimizing flow bridges between $\rho_0$ and $\rho_1$ via displacement interpolation (Facca et al., 2022, Papadakis et al., 2013).

Key extensions include:

• Riemannian/Manifold adaption: The continuity equation generalizes to manifolds by replacing $\rho_0$0 and $\rho_0$1 with metric and divergence from the Riemannian structure (Dong et al., 2024).
• Spatial–template and diffeomorphic constraints: Optimal transport regularization problems for imaging and registration couple the Benamou–Brenier action to ODE-constrained flows and Hilbert-space restrictions on admissible velocity fields (Chen, 2020).
• Inclusion of densities on graphs or multi-species labels: Vector-valued measures with graph mutation encoded via discrete Laplacians or couplings between multiple continua (Craig et al., 6 May 2025).
• Dynamic unbalanced transport (Wasserstein–Fisher–Rao): A quadratic source term in the continuity equation models mass creation/annihilation (Nishino et al., 10 Dec 2025, Gallouët et al., 2021).

2. Geometric and PDE Framework

Dynamic Benamou–Brenier formulations coincide with geodesics in the formal Riemannian metric induced by the Wasserstein structure on the space of probability measures. Specifically, solutions exhibit constant-action interpolation (displacement interpolation) and are governed by Hamilton–Jacobi equations in duality to the primal minimization.

• Metric property: For convex domains, the Benamou–Brenier action defines a true distance and admits a minimizing curve joining any pair of measures in $\rho_0$2 (Craig et al., 6 May 2025).
• Equivalence with Kantorovich: For compact domains, smooth Riemannian manifolds, or even sub-Riemannian manifolds without abnormal geodesics, dynamic and static (Kantorovich) distances coincide (Citti et al., 28 Jul 2025).
• Duality: Via the Fenchel–Rockafellar principle, the dual involves a Hamilton–Jacobi (HJ) type partial differential inequality for an appropriate potential function, with equality for strictly convex costs (Huesmann et al., 2017, Pass et al., 26 Sep 2025).

On Riemannian (and certain sub-Riemannian) manifolds, dynamic formulations generalize via replacement of $\rho_0$3 and $\rho_0$4 by the manifold gradient and metric. For $\rho_0$5 test functions, the PDE constraint writes

$\rho_0$6

and the action integrates $\rho_0$7 with respect to the volume form (Dong et al., 2024, Lavenant, 2019). Heat flow regularization contracts the Benamou–Brenier action by the exponential of Ricci curvature lower bounds (Bakry–Émery estimate) (Lavenant, 2019).

3. Algorithmic Approaches and Numerical Schemes

Dynamic Benamou–Brenier problems are discretized via (i) finite volumes or finite elements for the spatiotemporal grid, (ii) staggered or recovered gradients for spatial fluxes for accuracy and consistency on structured or unstructured meshes (Facca et al., 2022, Dong et al., 2024), and (iii) convex splitting algorithms, especially proximal splitting, ADMM, or interior-point methods.

Algorithmic highlights:

Algorithm	Key Feature	Scaling/Complexity
Interior point/Newton	Saddle-point reductions, Schur complement	BB-preconditioner, near-linear for large-scale (Facca et al., 2022)
Proximal splitting	Matrix-free, first-order updates	Fast for moderate accuracy and large problems (Papadakis et al., 2013)
ADMM (with gradient recovery)	Stable accuracy on triangulated surfaces	$\rho_0$8 error, mesh-unified (Dong et al., 2024)
Parallel Proximal Point (PPXA)	Unbalanced, constrained WFR	Block-splitting with O(1/k) convergence (Nishino et al., 10 Dec 2025)

Specialized preconditioners, such as the BB-preconditioner, are designed for saddle-point systems induced by Newton-KKT schemes, leveraging approximate operator commutation for dual Schur complements (Facca et al., 2022).

First-order splitting schemes, such as Douglas–Rachford (DR), Chambolle–Pock (CP), and ADMM, are deployed on staggered/centered grids, with pointwise proximal updates and global (often FFT-based) divergence solves, and support extensions to general costs, obstacles, and traffic-like constraints (Dong et al., 28 Jul 2025).

4. Dynamic Formulations Beyond Classical Wasserstein: Entropic, Unbalanced, Martingale, and Multi-Marginal Cases

Entropic (Schrödinger) case:

The entropic optimal transport problem admits a fluid-dynamical representation: $\rho_0$9 subject to a Fokker–Planck constraint $\rho_1$0, with unique solution given by the entropic interpolation. This extends the Benamou–Brenier formula to a convex action with Fisher information regularization and justifies the dynamic approach for measures with sub-Gaussian tails (Garatti et al., 28 Apr 2026, Gigli et al., 2018).

Unbalanced transport and WFR:

Dynamic unbalanced optimal transport introduces a source term in the continuity equation, penalized in action by Fisher–Rao (Hellinger–Kantorovich) cost: $\rho_1$1 with $\rho_1$2. Dynamic and static (entropy–transport) distances are equivalent, and the formulation supports analysis of Monge maps, polar factorization, Monge–Ampère equations for the WFR cost, and deduction of geometric regularity from the cost-convexity of the metric cone (Nishino et al., 10 Dec 2025, Gallouët et al., 2021).

Martingale optimal transport:

The martingale Benamou–Brenier problem minimizes the action

$\rho_1$3

among martingales with given initial and terminal (convex-order) marginals. The dynamic PDE is a Fokker–Planck equation (diffusion-only, no drift) and the theory aligns with classical BB via a stochastic–PDE correspondence (Huesmann et al., 2017, Backhoff-Veraguas et al., 2017, Guo et al., 26 Nov 2025).

Multi-marginal and vector-valued optimal transport:

Multi-marginal dynamic formulations use either coupled continuity equations or a single continuity equation on the product space, with translation-invariant or infimal-convolution cost. These dynamical problems are convex and recover Wasserstein barycenter formulations and quasi-Monge solutions (Krannich, 14 Dec 2025, Pass et al., 26 Sep 2025). For vector-valued measures, mass can mutate on a finite graph via label-mutation velocities, encoding competition or transformation between species and coupling the graph Laplacian structure explicitly in the PDE and action (Craig et al., 6 May 2025).

5. Extensions: Constraints, Couplings, and Physical Modeling

Dynamic Benamou–Brenier principles offer great modeling flexibility:

• Pointwise constraints: Saturation (e.g., traffic fundamental diagram), geometric obstacles, spacetime obstacles, and upper/lower bounds on density, flux, or source term are directly handled in the PDE constraints, typically as indicator functionals in proximal algorithms (Dong et al., 28 Jul 2025, Nishino et al., 10 Dec 2025).
• Integral, affine, or momentum constraints: Coupling conditions and generalized constraints (incl. inequalities) can be imposed for multi-class flows or resource-limited transport (Nishino et al., 10 Dec 2025).
• Stochasticity and random measures: The Benamou–Brenier principle also extends to curves of stationary random measures, with the action defined as an expectation under Palm probabilities and equivariant flows (Huesmann et al., 2024).

Hybrid formulations arise in imaging, inverse problems, and spatiotemporal regularization, where the BB energy serves as a dynamic regularizer within variational programs, and atomistic expansions and Frank–Wolfe-type algorithms exploit the extremal-curve characterization of dynamic optimal transport (Bredies et al., 2020).

6. Geometric, Physical, and Application-Centric Significance

The Benamou–Brenier framework unifies dynamic and static optimal transport on a geometric footing (Riemannian metric spaces, cones, sub-Riemannian geometry, Lorentzian spacetimes (Gigli et al., 19 Jan 2026)). It underpins the $\rho_1$4 formalism central to gradient flows, mean-field games, and dissipative PDEs.

• Stability and regularity: Dynamic and static distances are shown to be bi-Hölder equivalent for vector-valued settings, and the action functional admits strong regularity results even under mesh refinement in discrete settings (Craig et al., 6 May 2025, Lavenant, 2019).
• Gradient flows and PDE structure: The action metric-induced geometry yields canonical flows for interacting species and population models, and steepest descents compute mean-field evolution under interaction energies (Craig et al., 6 May 2025).
• Computational impact: Linearization techniques and lifted-space static distances (e.g. on product simplex spaces) accelerate computation in high-dimensional or large-sample regimes (Craig et al., 6 May 2025).

Practical applications span traffic modeling with congestion-aware constraints (Dong et al., 28 Jul 2025), population evolution with barriers or resources (Nishino et al., 10 Dec 2025), image registration with diffeomorphic transport (Chen, 2020), and robust reconstruction in spatiotemporal inverse problems (Bredies et al., 2020).

7. Limitations and Current Research Directions

While dynamic Benamou–Brenier formulations are broadly robust, challenges persist:

• Scalability: Interior-point methods face computational bottlenecks at very high resolution or as regularization parameters vanish ($\rho_1$5) (Facca et al., 2022).
• Parameter selection: First-order algorithms (e.g., DR, CP) require careful tuning of step sizes for fast convergence (Papadakis et al., 2013, Dong et al., 28 Jul 2025).
• Constraint complexity: High-order or nonlinear constraints (e.g., multi-class flows with complex phase space) can complicate proximal or augmented Lagrangian schemes (Nishino et al., 10 Dec 2025).
• Physical modeling: Extensions to higher-order traffic models (inertia, anticipation), stochastic effects, or network-level flow with nonlocal splitting remain open topics (Dong et al., 28 Jul 2025).
• Theoretical frontiers: Analysis of the Monge–Ampère PDE under WFR on Riemannian manifolds, sharp regularity-transfer results (via metric cones), and extensions to Lorentzian or noncommutative geometries are under current study (Gallouët et al., 2021, Gigli et al., 19 Jan 2026).

The dynamic Benamou–Brenier paradigm thus continues to serve as a central, flexible, and extensible framework in the theory and computation of optimal transport and its many generalizations.