Optimal Transport Flows

• Optimal Transport Flows are dynamical systems that continuously transport mass by minimizing kinetic energy under convex and capacity constraints.
• They integrate dynamic formulations, traffic constraints, and convex discretization to yield scalable optimization methods for congestion-aware network models.
• The framework unifies theoretical analysis and practical implementations, offering rigorous convergence guarantees and extensions to neural ODEs and adaptive networks.

Optimal transport flows are dynamical systems that realize optimal transport couplings or maps as time-continuous flows of mass (or probability), minimizing an action that corresponds to transport cost, and generalizing both static optimal transport and classical fluid dynamics or traffic flow models. This concept unifies a wide range of mathematical, computational, and application-focused frameworks across the physical sciences, engineering, and machine learning. Advances in optimal transport flows span convex variational principles, monotone operator theory, fundamental-diagram-constrained dynamical systems on graphs, Hamilton–Jacobi and neural ODE solvers, and large-scale convex optimization for real-world transportation networks.

1. Dynamic Formulation: Benamou–Brenier and Kinetic Action Frameworks

The prototypical mathematical formulation of an optimal transport flow is the dynamic (Benamou–Brenier) problem: $$\min_{\rho,\,m} \int_0^1 \int_{\Omega} \frac{\|m(t,x)\|^2}{2\rho(t,x)}\,dx\,dt,\quad \text{subject to}\;\; \partial_t\rho+\nabla\cdot m=0,\,\rho(0)=\rho_0,\,\rho(1)=\rho_1,$$ where $\rho(t,x)$ is a time-dependent density, $m(t,x)$ is the associated momentum or flux field, and the action functional encodes the total kinetic energy of mass transported from $\rho_0$ to $\rho_1$. In the network/graph setting, densities reside on nodes and momenta on edges, connected via a discrete continuity equation involving the graph incidence operator (Dong et al., 1 Nov 2025).

The energetic cost is typically expressed as a sum of perspective functions $m^2/(2\rho)$, which is strictly convex for positive density. This perspective is crucial when passing to discretizations and constructing provably convergent numerical algorithms.

2. Structure, Semantics, and Constraints for Traffic and Network Flows

In applications such as traffic or computational logistics, graph-based formulations place special significance on modeling physical constraints and interpreting variables:

• Density $\rho$ is stored at nodes (intersections/accumulation points).
• Momentum/flux $m$ is assigned to directed edges (road segments/capacity links), matching real-world semantics (Dong et al., 1 Nov 2025).
• Network dynamics obey a discrete mass conservation ODE: $\dot \rho = D^\top m$.

A pivotal innovation is the incorporation of fundamental diagram (FD) constraints. Here, each edge $e$ receives a density-dependent capacity bound $Q_e(\rho)$ (e.g., Greenshields' relation $Q(\rho) = v_0 \rho (1-\rho/\hat\rho)$), so that flux is capped: $0 \leq m(e) \leq Q_e(\bar{\rho}_e)$. The resulting optimization is convex due to the concavity of $Q$ in $\rho$ and is both mass-conserving and congestion-aware, a direct analogy to macroscopic traffic models (e.g., Lighthill–Whitham–Richards) (Dong et al., 1 Nov 2025, Dong et al., 28 Jul 2025).

3. Convex Discretization and Numerical Solvers

The continuous problem is discretized in time and/or space for tractable computation:

• Time discretization introduces steps $\rho_i$ (densities) and $m_i$ (momentum), with mass-balance enforced at each step (Dong et al., 1 Nov 2025).
• The fundamental-diagram constraint is imposed at each time/edge by evaluating midpoint densities and projecting momenta accordingly.
• The full discrete optimization is a convex program with block structure, enabling scalable algorithms with theoretical guarantees.

A widely used numerical scheme is ADMM, which splits the variables to isolate difficult constraints and iteratively solves convex subproblems—often reducing to sparse linear algebra (e.g., block-tridiagonal systems for $\rho$, projections for $q$, quadratic solves for $m$) (Dong et al., 1 Nov 2025). Proximal splitting methods (Douglas–Rachford, Chambolle–Pock) also exploit the separable structure and affine constraints, yielding mesh-agnostic, scalable solvers for high-dimensional and geometric domains (Dong et al., 28 Jul 2025).

Table: Core Structural Components in FD-Constrained OT Flows (Graph-Based)

Concept	Variable Placement	Physical Interpretation
Density ($\rho$)	Nodes	Vehicle accumulation at intersections
Momentum/Flux ($m$)	Edges	Flow along road segments
Capacity ($Q_e$)	Edges (FD constraint)	Max flux by local density (traffic physics)

4. Existence, Uniqueness, and Theoretical Guarantees

The strict convexity of the kinetic cost and the convex constraint set jointly ensure existence and uniqueness of optimal transport flows with capacity constraints—conditional only on feasibility of the mass-balance and FD inequalities. Uniqueness is proved by showing that any two solutions can be strictly convexly averaged to reduce action unless they coincide everywhere. Consequently, global convergence can be established for the iterative methods described above (Dong et al., 1 Nov 2025).

5. Empirical and Application Results

Numerical experiments on both synthetic and real urban networks demonstrate essential phenomena:

• In simple line graphs, mass is displaced smoothly, with flow thickening along high-flux edges, and conservation apparent at all times (Dong et al., 1 Nov 2025).
• On city-scale networks (e.g., Athens), congestion-aware OT flows split over parallel routes to avoid bottlenecks, and capacity constraints produce more realistic (lower, distributed) peak fluxes than unconstrained OT, which tends to overload shortest paths.
• The algorithmic splitting and block structure yield empirical scalability: per-iteration complexity is dominated by sparse solves, and total iteration counts are modest (hundreds for high-precision primal/dual convergence) (Dong et al., 1 Nov 2025).
• The methodology supports flexible incorporation of obstacles, time-varying capacities, or sources/sinks, making it adaptable for real-world design tasks in mobility and logistics (Dong et al., 28 Jul 2025).

6. Traffic-Theoretic and Modeling Implications

Fundamental-diagram-constrained OT flows generalize classical dynamic OT (Benamou–Brenier) by embedding physical rate-limiting mechanisms into the feasible set:

• Recover classical traffic models such as LWR (where $m=Q(\rho)$ always) in the equilibrium limit; the relaxed $m \leq Q(\rho)$ setting supports non-equilibrium and transitionary regimes endemic in practical networks (Dong et al., 28 Jul 2025).
• Enable seamless modeling of "spillback", rerouting, and capacity-driven mass redistribution—critical for congestion-sensitive planning and origin-destination analysis (Dong et al., 1 Nov 2025, Ibrahim et al., 2021).
• The framework admits generalizations to multi-modal, multi-class flows, temporally and spatially varying constraints, and hierarchical (multi-layer) network structures.

7. Extensions, Limitations, and Future Directions

The convex variational structure of these optimal transport flows under capacity constraints supports a broad research frontier:

• Integration with neural network parameterizations of the flow, embedding learned cost metrics or demand structure (Xu et al., 2023, Huguet et al., 2022).
• Applications in non-transport domains, such as adaptive mesh refinement (via Monge–Ampère) or model reduction for hyperbolic PDEs, exhibit the universal modeling capacity of optimal transport flows (Heyningen et al., 2023).
• Open challenges include extending theoretical guarantees to nonconvex or stochastic settings, scalable distributed algorithms for massive graphs, and statistical learning of empirical fundamental diagrams from real-world traffic data.

The traffic-constrained optimal transport flow paradigm thus unifies classical transport, network optimization, and kinetic physics, yielding a mature and powerful methodology for both theoretical analysis and practical algorithmic deployment in large-scale, capacity-limited mass transport systems (Dong et al., 1 Nov 2025, Dong et al., 28 Jul 2025, Ibrahim et al., 2021).