Dynamic Unbalanced Optimal Transport

• Dynamic unbalanced OT is defined by a continuity equation with a source term that fuses spatial mass transport with local mass creation or destruction.
• It employs diverse variational formulations—dynamic, static, convex, and nonconvex—that balance transport costs with penalties for mass change.
• Scalable algorithms using convex optimization and neural architectures facilitate efficient high-dimensional applications in biology, chemistry, and population dynamics.

Dynamic unbalanced optimal transport (OT) extends the dynamic Benamou–Brenier paradigm of optimal transport by allowing both spatial mass transfer and mass variation (creation or destruction) along transport paths. This modeling framework is essential in situations where conservation of mass is not guaranteed, such as biological growth, chemical reactions, and population dynamics. The theory encompasses a diverse range of variational formulations—dynamic and static, continuous and discrete, convex and non-convex—each tailored to applications and computational demands. Central to the dynamic unbalanced OT framework are (i) the specification of a continuity equation with a source term, (ii) variational functionals penalizing both transport and mass change, (iii) a duality theory connecting dynamic and static problems, and (iv) scalable algorithms involving convex optimization, neural architectures, and control-theoretic reductions.

1. Dynamic Unbalanced Optimal Transport: Mathematical Formulation

Dynamic unbalanced OT generalizes the classical Benamou–Brenier formulation by introducing mass density $\rho_t(x)\in \mathcal{M}(\Omega)$, mass flux $j_t(x)\in \mathcal{M}(\Omega)^n$, and a mass change (source/sink) rate $s_t(x)\in \mathcal{M}(\Omega)$ for $t\in[0,1]$. The evolution is governed by a continuity equation with source: $$\partial_t \rho_t(x) + \nabla\cdot j_t(x) = s_t(x),\quad \rho_{t=0} = \rho_0, \quad \rho_{t=1} = \rho_1$$ (where Neumann boundary conditions are often imposed at $\partial\Omega$). The canonical unbalanced OT problem, with Wasserstein-1 transport and a convex 1-homogeneous mass-change cost $C_D$, is to minimize

$$P_D(\rho, j, s) = \int_0^1\left[\|j_t\|_{TV} + \int_\Omega C_D(\rho_t(x), s_t(x))\, dx\right]\,dt$$

with $\|j_t\|_{TV} = \int_\Omega |j_t(x)|\,dx$ and $C_D$ satisfying $j_t(x)\in \mathcal{M}(\Omega)^n$0 if $j_t(x)\in \mathcal{M}(\Omega)^n$1, $j_t(x)\in \mathcal{M}(\Omega)^n$2 if $j_t(x)\in \mathcal{M}(\Omega)^n$3 (Schmitzer et al., 2017).

Alternative dynamic formulations penalize both displacement and mass-change via a quadratic structure. In Wasserstein–Fisher–Rao (WFR) geometry,

$j_t(x)\in \mathcal{M}(\Omega)^n$4

subject to

$j_t(x)\in \mathcal{M}(\Omega)^n$5

with $j_t(x)\in \mathcal{M}(\Omega)^n$6 controlling local production ($j_t(x)\in \mathcal{M}(\Omega)^n$7) or annihilation ($j_t(x)\in \mathcal{M}(\Omega)^n$8) of mass (Wan et al., 2024, Peng et al., 11 Jan 2026).

2. Duality and Static Reductions

A remarkable structural result, particularly for Wasserstein-1–type costs, is that the dynamic unbalanced OT problem admits a static dual and primal formulation through convex duality. The dynamic dual: $j_t(x)\in \mathcal{M}(\Omega)^n$9 subject to $s_t(x)\in \mathcal{M}(\Omega)$0 and $s_t(x)\in \mathcal{M}(\Omega)$1, is shown (via Fenchel–Rockafellar duality) to be equivalent to the static dual: $s_t(x)\in \mathcal{M}(\Omega)$2 where $s_t(x)\in \mathcal{M}(\Omega)$3 is derived from $s_t(x)\in \mathcal{M}(\Omega)$4 by time integration. Conjugation transposes this to a static semi-coupling problem involving measures $s_t(x)\in \mathcal{M}(\Omega)$5, $s_t(x)\in \mathcal{M}(\Omega)$6 on $s_t(x)\in \mathcal{M}(\Omega)$7 (Schmitzer et al., 2017).

This equivalence enables the reformulation of the original time-dependent variational problem into one over static objects, facilitating efficient numerical solution schemes such as linear programming, primal–dual splitting, or entropic regularization.

3. Structure and Characterization of Dynamic Unbalanced OT Solutions

Optimal solutions in dynamic unbalanced OT exhibit a distinctive structure:

• All spatial transport occurs at the endpoints $s_t(x)\in \mathcal{M}(\Omega)$8 and $s_t(x)\in \mathcal{M}(\Omega)$9, with $t\in[0,1]$0 and $t\in[0,1]$1 representing mass flows.
• For intermediate times $t\in[0,1]$2, only pointwise mass changes occur; no spatial redistribution is present.
• The optimizer thus decomposes into initial transport, purely local in-place mass evolution, and final transport.

In one-dimensional Dirac examples ($t\in[0,1]$3, $t\in[0,1]$4), the optimal partition between transport and in-situ mass modification is characterized by geometric tangent-line conditions imposed by the mass-change cost function $t\in[0,1]$5 (Schmitzer et al., 2017).

A consequence is the existence of a maximal threshold distance $t\in[0,1]$6 such that mass is only relocated over distances $t\in[0,1]$7, beyond which mass creation or destruction is favored energetically.

4. Computational Methods and High-dimensional Extensions

Dynamic unbalanced OT's static reduction supports efficient high-dimensional optimization. For general Wasserstein–Fisher–Rao models,

• Lagrangian discretization tracks characteristic curves, with densities evolved via coupled ODEs and Monte Carlo integration over initial samples for scalability (Wan et al., 2024).
• Neural-parameterized velocity and source fields enable flexible modeling of the transport and mass-creation vectors and source functions, with automatic differentiation for gradient computation and stochastic gradient optimization (e.g., Adam).
• Approaches on surfaces leverage mesh-free neural representations with structure-exploiting physics-informed losses derived from the underlying Hamiltonian flows of the KKT system (Pan et al., 2024).

In control-theoretic and discrete-time settings, for Gaussian reference measures and linear dynamics, the infinite-dimensional dynamic UOT problem admits reduction to a convex semidefinite program (SDP) over moments and covariances, with closed-form updates for optimal transported mass (Nakashima et al., 5 May 2026, Nakashima et al., 7 May 2026).

Simulation-free learning of unbalanced OT geodesics is tractable through flow matching, wherein neural networks directly regress both velocity and mass-change fields against analytically constructed targets, bypassing ODE solvers and leading to substantial empirical speedup and improved accuracy (Peng et al., 11 Jan 2026).

5. Generalizations, Constraints, and Synchronization

Dynamic unbalanced OT readily accommodates integral path constraints, such as:

• Fixed total mass curves, prescribed statistical moments, or more general affine functionals of densities and fluxes (Bauer et al., 2024, Nishino et al., 10 Dec 2025).
• Inequality and equality constraints on densities, fluxes, or source terms.
• Constraints can be imposed in the static reduction or directly in the time-dependent variational problems, preserving convexity and the existence of minimizers.

Synchronization across multiple spaces (e.g., modalities or geometric domains) is formalized through Unbalanced Synchronized OT, which couples transport-reaction flows either via Monge (push-forward) or Kantorovich (Markov kernel) synchronization operators. In the Monge case, transport costs are metric–modified Benamou–Brenier actions; in the Kantorovich case, nonlocal dissipation actions characterize the joint evolution. These extensions admit efficient discretizations and primal–dual solution schemes (Cang et al., 21 Feb 2026).

6. Special Cases and Connections to Classical OT

Dynamic unbalanced OT frameworks recover standard mass-preserving OT in the zero-mass-change limit (e.g., $t\in[0,1]$8 for $t\in[0,1]$9, $$\partial_t \rho_t(x) + \nabla\cdot j_t(x) = s_t(x),\quad \rho_{t=0} = \rho_0, \quad \rho_{t=1} = \rho_1$$0 otherwise), realizing the Benamou–Brenier $$\partial_t \rho_t(x) + \nabla\cdot j_t(x) = s_t(x),\quad \rho_{t=0} = \rho_0, \quad \rho_{t=1} = \rho_1$$1 or $$\partial_t \rho_t(x) + \nabla\cdot j_t(x) = s_t(x),\quad \rho_{t=0} = \rho_0, \quad \rho_{t=1} = \rho_1$$2 geodesics (Schmitzer et al., 2017). The quadratic class (Wasserstein–Fisher–Rao or Hellinger–Kantorovich) blends displacement and Fisher–Rao mass creation costs, resulting in joint penalization and admitting both dynamic and static (semi-coupling, “soft marginal”) formulations.

Partial transport and total variation penalized marginals are subsumed by appropriate choices of $$\partial_t \rho_t(x) + \nabla\cdot j_t(x) = s_t(x),\quad \rho_{t=0} = \rho_0, \quad \rho_{t=1} = \rho_1$$3, and are represented naturally within this unifying framework.

7. Applications and Extensions

Dynamic unbalanced OT has been applied in:

• High-dimensional distribution interpolation and density matching under mass-creation and loss (e.g., computational biology, population flows, density control in engineered systems) (Peng et al., 11 Jan 2026, Wan et al., 2024).
• Shape morphing and registration on surfaces or point clouds, robustly handling noisy geometric data (Pan et al., 2024).
• Synchronized transport across heterogeneous modalities in multi-view data (Cang et al., 21 Feb 2026).
• Constrained interpolation under prescribed statistical or geometric constraints (e.g., convex body morphing, trajectory control with resource budgets) (Bauer et al., 2024, Nishino et al., 10 Dec 2025).

Algorithmic innovations—scalable convex formulations, parallel proximal splitting, simulation-free flow matching—combine with neural parameterization to facilitate efficient solution schemes, handling previously intractable high-dimensional or large-scale instances. The incorporation of affine and integral path constraints broadens the range of admissible applications while maintaining mathematical tractability.

---

The theory of dynamic unbalanced optimal transport now supports a broad spectrum of variational, duality, computational, and modeling generalizations, with precise correspondence between dynamic time-evolution problems and static optimization formulations, robust structural results for optimizer characterizations, and a diverse ecosystem of scalable, structure-exploiting solvers (Schmitzer et al., 2017, Wan et al., 2024, Pan et al., 2024, Peng et al., 11 Jan 2026, Wu et al., 4 Apr 2025, Nishino et al., 10 Dec 2025, Bauer et al., 2024, Cang et al., 21 Feb 2026, Nakashima et al., 5 May 2026, Nakashima et al., 7 May 2026).