Lagrangian Vector Fields: Theory & Applications

• Lagrangian vector fields are defined as the velocity fields generating mass-preserving flows on Riemannian manifolds, linking particle trajectories with evolving densities.
• They underpin optimal transport by formulating geodesic flows via the Benamou–Brenier approach, ensuring continuity and energy minimization.
• These vector fields are applied in image morphing, fluid dynamics, and machine learning, with numerical schemes ensuring accurate discretization and regularity.

A Lagrangian vector field is a fundamental construct in differential geometry, optimal transport, and the analysis of mass-conserving flows. It serves as the infinitesimal generator of Lagrangian (mass-preserving) flows, encoding the velocity field along which a distribution (or a set of particles) is transported in time. The rigorous analysis and computation of Lagrangian vector fields underpin the dynamic (Eulerian and Lagrangian) formulations of optimal transport and continuum mechanics.

1. Mathematical Definition and Context

Let $(M, g)$ be a smooth $n$-dimensional Riemannian manifold, and let $\rho(x, t)$ denote a time-dependent nonnegative density with $\int_M \rho(x, t)\,dx = 1$. A vector field $v(x, t) \in TM$ is termed a Lagrangian vector field if it generates the flow

$$\frac{d}{dt} X_t(x) = v(X_t(x), t), \quad X_0(x) = x,$$

such that the push-forward of the initial density by the flow $X_t$ reproduces the evolving density: $(X_t)_\# \rho_0 = \rho_t.$ This ensures mass conservation (the mapping is measure-preserving) and links the Lagrangian perspective (following trajectories) to the Eulerian description (evolution of the density field).

The associated continuity equation is

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0,$$

ensuring that the vector field $v$ is compatible with the instantaneous mass distribution. In the context of optimal transport, the Lagrangian vector field yields the geodesics of the Wasserstein metric on the space of probability measures (Vandegriffe, 2020, Solomon, 2018).

2. Lagrangian Vector Fields in Optimal Transport

Optimal transport theory provides a direct link between Lagrangian vector fields and measure-theoretic geodesics. Given two probability densities $\rho_0$ and $\rho_1$ on $\Omega\subset\mathbb{R}^n$, the Benamou–Brenier dynamic OT formulation seeks the vector field $v(x,t)$ minimizing the kinetic energy under the continuity equation constraint: $$\inf_{\rho,v} \int_0^1 \int_\Omega \frac{1}{2} |v(x,t)|^2 \rho(x,t)\, dx \,dt,$$ subject to $\partial_t \rho + \nabla \cdot (\rho v)=0$, $\rho(\cdot,0) = \rho_0$, $\rho(\cdot,1) = \rho_1$ (Vandegriffe, 2020, Solomon, 2018). The optimal vector field $v^*(x, t)$ is a time-dependent Lagrangian vector field and yields the geodesic flow connecting $\rho_0$ to $\rho_1$ in Wasserstein space.

A critical property is that for $L^2$-Wasserstein OT on $\mathbb{R}^n$, the velocity field $v(x, t) = \nabla \varphi(x, t)$ is gradient, and $\varphi(x, t)$ evolves following the Hamilton–Jacobi equation. For distributions with smooth strictly positive densities, the optimal flow is given by the gradient of a convex potential interpolating between initial and target maps.

3. Structure and Regularity

The regularity of Lagrangian vector fields is governed by the geometry of the underlying space and the nature of the measure evolution. For quadratic cost, Brenier’s theorem guarantees that the optimal transport is given by a gradient of a convex function, and so the Lagrangian vector field is globally integrable and smooth if the source and target are absolutely continuous measures with smooth densities (Lindsey et al., 2016).

For more general costs or under constraints (such as incompressibility or pointwise bounds on density/flux as in crowd motion and capacity-constrained transport), the vector field may be non-smooth or only in a weak (Sobolev or measure-theoretic) sense (Kerrache et al., 2022). In such cases, existence of Lagrangian flows (in the sense of DiPerna–Lions or Ambrosio) remains possible under certain conditions on the vector field’s regularity and the continuity equation’s well-posedness.

4. Discretization and Numerical Construction

In computational optimal transport, Lagrangian vector fields are approximated via variational or PDE-based schemes. For instance, finite-dimensional discretizations of the dynamic OT problem yield piecewise-constant or affine approximations to $v(x, t)$, ensuring that the discrete continuity equation holds (Lindsey et al., 2016). In the so-called Benamou–Brenier discretization, the velocity field is represented on a space–time grid or mesh, and the optimization is performed over $(\rho, v)$ pairs satisfying the finite difference version of the continuity equation.

A formal summary:

Numerical Scheme	Representation of $v(x,t)$	Comments
Benamou–Brenier (PDE-based)	Grid-based, time-dependent field	Convex optimization in $(\rho,v)$
Monge–Ampère Discretization	Implicit in gradient of convex potential	Applies for quadratic cost
Graph-based/OT on Graphs	Flow along edges	Pseudo-time Lagrangian vector field (Grover et al., 2016)

In PDE-inspired methods, the optimal Lagrangian vector field induces the displacement interpolation, and the discretized flow fields can be visualized as trajectories moving mass from the initial to the final configuration.

5. Role in Geometry and Physics

The Lagrangian vector field formalizes the concept of "mass transporters" in several geometric and physical frameworks. In the Wasserstein geometry, it describes geodesics on the infinite-dimensional Riemannian manifold of probability densities (McCann, 2012). In fluid dynamics and incompressible flow, it matches the notion of the material velocity field; in differential geometry, Lagrangian vector fields are associated with Lagrangian submanifolds (i.e., submanifolds along which a symplectic form vanishes).

For constrained mass transport (e.g., with upper/lower bounds on $\rho(x,t)$ or $\|v(x,t)\|$), the feasible space of Lagrangian vector fields is further restricted to those obeying the required pointwise or integral inequalities (Kerrache et al., 2022). Algorithmic frameworks exploit proximal operators and convex splitting methods to construct discrete Lagrangian vector fields that respect these constraints.

6. Applications and Extensions

Lagrangian vector fields are exploited in:

• Image morphing and interpolation: mapping images or densities along optimal paths by solving for the underlying Lagrangian vector field between given data (Martín et al., 19 Jun 2024).
• Graph-based transport and mixing: designing vector fields on graphs (discrete state spaces) for mixing control or measure transport (Grover et al., 2016).
• Constrained physical flows: crowd motion, granular flows, and geophysical models utilize Lagrangian vector fields under mass, momentum, or density constraints (Kerrache et al., 2022).
• Machine learning: modeling distribution morphing or generative modeling by parameterizing the Lagrangian vector field via neural networks (Korotin et al., 2022, Martín et al., 19 Jun 2024).

A plausible implication is that extensions to non-Euclidean, Riemannian, or discrete manifold settings rely fundamentally on constructing Lagrangian vector fields compatible with the geometry and structure of the space (Tupitsa et al., 2022).

7. Connections to Analytical and Geometric Structures

The existence, uniqueness, and qualitative properties of Lagrangian vector fields are tightly coupled to the differential-topological structure of optimal maps and transport plans. For strictly convex costs, and under certain geometrical conditions (e.g., Ma–Trudinger–Wang curvature), regularity theory guarantees smoothness and stability of the Lagrangian flows (McCann, 2012). For general metric-measure spaces, the stability of Lagrangian vector fields under weak convergence of the marginals is ensured by tightness and lower semicontinuity of the associated cost functionals (Vandegriffe, 2020).

Lagrangian vector fields also formalize the velocity field along which probability measures move in the tangent space of Wasserstein space, aligning with Otto's formal Riemannian calculus on the space of measures—a structure central in the modern theory of metric-measure geometry and optimal transport (McCann, 2012, Solomon, 2018).