Constrained Mass Transport (CMT)

• Constrained mass transport (CMT) is a family of optimal transport models that impose extra restrictions—such as capacity limits, affine constraints, and source mechanisms—beyond simple mass conservation.
• CMT frameworks span diverse settings, including excluded-volume effects in lattice gases, path restrictions in unbalanced models, and capacity constraints in traffic and network flows.
• These models build on the Benamou–Brenier formulation and employ variational methods, proximal splitting, and augmented Lagrangian techniques to optimize mass redistribution within a reduced feasible class.

Constrained mass transport (CMT) denotes transport models in which admissible mass evolution is restricted by additional structure beyond the standard continuity equation and endpoint conditions of optimal transport. Across the literature, these restrictions take several non-equivalent forms: excluded-volume coupling in an inhomogeneous medium, affine constraints on density or flux paths, source/sink mechanisms in unbalanced transport, flow-rate or capacity limits on routes and networks, prescribed zero patterns in discrete transport plans, and stepwise information-theoretic constraints in probability space. The literature therefore suggests that CMT is best understood not as a single canonical model but as a family of constrained transport paradigms in which redistribution is optimized over a reduced feasible class rather than over all mass-preserving motions (Lukyanets et al., 2010, Kerrache et al., 2022, Bauer et al., 2024, Klitzing et al., 21 Oct 2025).

1. Standard transport baseline and the meaning of “constraint”

The common baseline for many CMT formulations is the dynamic Benamou–Brenier formulation of balanced optimal transport. In its standard quadratic form, one minimizes

$$\min_{\rho,m}\int_0^1\int_\Omega \frac{|m(t,x)|^2}{2\rho(t,x)}\,dx\,dt$$

subject to

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$

This formulation expresses transport as a kinetic-energy minimization over density and momentum fields. In the unconstrained setting, the only hard structure is mass conservation together with the endpoint marginals (Kerrache et al., 2022).

CMT arises when additional admissibility conditions are imposed on this dynamic picture or on an equivalent static formulation. In the abstract Hilbert-space formulation of constrained optimal transport, the transport problem is recast as a saddle-point system with convex functionals and a linear coupling constraint $B\phi=q$, and the constrained problem is defined either by imposing a soft constraint on density and momentum fields or by restricting the trajectory to a prescribed subset (Kerrache et al., 2022). In the path-constrained unbalanced setting, admissible paths are required to satisfy affine integral conditions such as

$\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$

thereby constraining the entire measure path rather than only its endpoints (Bauer et al., 2024). In discrete transport, the constraint may instead act directly on the coupling support, for example through prior-imposed zeroes $t_{ij}=0$ for forbidden source–target pairs (Corless et al., 2024). In traffic and network formulations, admissibility is encoded by capacity laws on edges rather than by geometric freedom alone (Dong et al., 2022, Dong et al., 1 Nov 2025).

A recurrent misconception is that CMT is synonymous with unbalanced optimal transport. The literature does not support that identification. Some constrained models remain mass-preserving and instead restrict support, boundary images, path sets, or edge fluxes; others relax conservation by introducing source terms while imposing new constraints on where, when, or how mass variation may occur (Zhu et al., 2020, Chen et al., 2023, Bauer et al., 2024).

2. Excluded-volume CMT in inhomogeneous media

A distinct and earlier use of the term appears in the study of a two-component lattice gas with excluded volume, where constrained mass transport refers to motion of a mobile component through an inhomogeneous medium under the hard occupancy constraint

$\hat m_i+\hat n_i\le 1.$

Each site can hold at most one particle total, the two species are distinguishable, and nearest-neighbor hopping occurs with rates $\nu_m,\nu_n$. In the long-wavelength approximation, the continuum equations become

$$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$

with mixing flux

$j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$

This mixing term is absent in ordinary one-component diffusion and is interpreted as mutual drag generated by distinguishability together with hard-core repulsion (Lukyanets et al., 2010).

The CMT regime in that paper is the frozen-background limit $\nu_n\ll \nu_m$, in which the $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$0-component is treated as a quenched inhomogeneous field. The mobile density then satisfies

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$1

with flux

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$2

The field $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$3 therefore enters both as a reduced local diffusivity $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$4 and as a drift/compression term $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$5. This is the core mathematical statement of CMT in that setting (Lukyanets et al., 2010).

Several transport effects follow. First, density relaxation can be locally accompanied by compression: near minima of the frozen field, the mobile component accumulates, while near maxima it is expelled. The short-time estimates

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$6

quantify this local growth and decay. Second, in quasi-one-dimensional periodic backgrounds, a Gaussian packet does not spread smoothly but crosses barriers in stages, yielding a hopping-like, piecewise-linear front dynamics with alternating slow and fast phases. Third, the packet can fragment into subpackets, so the usual root-mean-square displacement

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$7

becomes a misleading transport diagnostic because it averages over trapped, spreading, and newly crossed fragments simultaneously (Lukyanets et al., 2010).

The stationary problem is likewise nonclassical. For

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$8

the total flux is

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$9

Hence flux direction depends jointly on $B\phi=q$0 and $B\phi=q$1 at the boundaries. In particular, if

$B\phi=q$2

transport is directed from the lower-$B\phi=q$3 boundary toward the higher-$B\phi=q$4 boundary. The paper explicitly attributes these effects to distinguishability plus excluded volume; in an indistinguishable one-component gas, the mixing flux is absent and the comparable CMT effect does not occur (Lukyanets et al., 2010).

3. Dynamic optimal-transport CMT: unbalanced, path-constrained, and source-augmented models

A major line of work interprets CMT within dynamic optimal transport by retaining the variational transport structure while constraining mass balance, admissible paths, or mass variation mechanisms. In unbalanced formulations, conservation is relaxed by introducing a source term. A Fisher–Rao-type dynamic model replaces

$B\phi=q$5

with

$B\phi=q$6

and minimizes

$B\phi=q$7

Here positive $B\phi=q$8 creates mass, negative $B\phi=q$9 removes it, and $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$0 controls the relative cost of source usage (Zhu et al., 2020).

That source mechanism can be embedded into vector-valued optimal transport. The key construction introduces an additional “source layer” and interprets the scalar source as inter-channel flow in an augmented graph: $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$1 With two channels $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$2 and layer weights $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$3, $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$4 very small, the weighted vector-valued cost suppresses unwanted energy on the source layer, so unbalanced OMT is approximated by vector-valued OMT on an augmented graph with a lightly weighted source layer. The same idea extends to unbalanced vector-valued OMT by connecting the source layer to each existing channel through additional edges weighted by $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$5 (Zhu et al., 2020).

Regularized variants add diffusion. In unbalanced regularized optimal mass transport, the continuity equation becomes

$\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$6

and the action is

$\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$7

The variable $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$8 is a relative source, $\int_\Omega H(t,x)\,d\rho_t(x)=F(t),$9 is an indicator activating the source only where allowed, and $t_{ij}=0$0 controls the trade-off between transport and source-mediated adjustment. The paper explicitly interprets the source term as Fisher–Rao and describes the model as a two-channel picture comprising physical-space transport and exchange with an “invisible sink/source layer.” As $t_{ij}=0$1, the source is suppressed and the model approaches balanced rOMT (Chen et al., 2023).

Path-constrained unbalanced OT generalizes this further by restricting the intermediate measures themselves. In the WFR setting, one minimizes

$t_{ij}=0$2

while requiring $t_{ij}=0$3 to satisfy affine constraints such as

$t_{ij}=0$4

The framework covers total-mass constraints, prescribed time-dependent mass, moment constraints, moving barriers, probability-density constraints, and closure constraints for area measures. For suitable classes of constraints, existence of minimizers is established; under the assumption that every pair of constrained measures can be connected by a finite-energy path, the induced distance is a metric and arises as the geodesic distance of a Riemannian metric through an analogue of Otto’s submersion (Bauer et al., 2024).

A later extension allows affine equality and inequality constraints to act not only on density but also on momentum and source terms. The general constraint takes the form

$t_{ij}=0$5

or equality instead of inequality. This encompasses standard balanced OT, standard WFR, earlier density-only constrained models, spherical Hellinger–Kantorovich, directional momentum constraints, and source-budget constraints within a single convex framework (Nishino et al., 10 Dec 2025).

A related $t_{ij}=0$6-type unbalanced theory shows that linear-in-distance transport cost yields a particularly rigid constrained structure. The dynamic model

$t_{ij}=0$7

with 1-homogeneous flux penalty is equivalent to a computationally more efficient static model. Under a mild condition on the mass-change function, dynamic optimizers split into instantaneous transport at $t_{ij}=0$8, pure mass change on $t_{ij}=0$9, and instantaneous transport at $\hat m_i+\hat n_i\le 1.$0, and not every static discrepancy admits a dynamic realization. This rules out the assumption that all constrained unbalanced formulations are dynamically equivalent (Schmitzer et al., 2017).

4. Capacity, topology, and boundary constraints

Another major branch of CMT concerns admissible transport routes and throughput limits. In transport through constrictions, one considers moving unit mass from $\hat m_i+\hat n_i\le 1.$1 to $\hat m_i+\hat n_i\le 1.$2 while forcing trajectories to pass through one or more tolls at specified points. For a single toll at $\hat m_i+\hat n_i\le 1.$3, the unknown is a coupling $\hat m_i+\hat n_i\le 1.$4, where $\hat m_i+\hat n_i\le 1.$5 is the crossing time. The flow-rate constraint is imposed on the time marginal: $\hat m_i+\hat n_i\le 1.$6 and the kinetic cost is

$\hat m_i+\hat n_i\le 1.$7

This casts transportation scheduling as a generalized multi-marginal Kantorovich problem in which passage time through the constriction becomes an optimization variable. Existence holds under $\hat m_i+\hat n_i\le 1.$8, and in a one-dimensional separated-support configuration the minimizer is unique and induced by monotone rearrangement (Dong et al., 2022).

Capacity constraints also appear in ramified transport. Instead of a single transport current, the constrained model uses a transport multi-path

$\hat m_i+\hat n_i\le 1.$9

with each component carrying at most mass $\nu_m,\nu_n$0: $\nu_m,\nu_n$1 The cost is the sum of $\nu_m,\nu_n$2-costs,

$\nu_m,\nu_n$3

and for $\nu_m,\nu_n$4 an optimal multi-path exists with finitely many components satisfying

$\nu_m,\nu_n$5

Each component is itself an optimal transport path, and in the atomic case all but at most $\nu_m,\nu_n$6 components are map-compatible. The model therefore preserves much of branched transport geometry while forbidding unlimited aggregation into a single high-capacity trunk (Xia et al., 2024).

On graphs, dynamic OT can be constrained by traffic-theoretic capacity laws. In the fundamental-diagram-constrained formulation, densities live on nodes, momentum on directed edges, and the continuity equation is

$\nu_m,\nu_n$7

The kinetic action

$\nu_m,\nu_n$8

is supplemented by edgewise capacity constraints

$\nu_m,\nu_n$9

with Greenshields law

$$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$0

Under a feasibility assumption, the optimal flow is unique. The model remains OT-like in its variational structure but becomes congestion-aware and capacity-limited (Dong et al., 1 Nov 2025).

Boundary constraints represent a different geometry of admissibility. In the Monge–Ampère approach to $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$1 optimal transport, the transport map $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$2 must satisfy the second boundary value condition

$$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$3

A practical numerical realization solves a sequence of Monge–Ampère equations with Neumann conditions, updating boundary data by projecting the current boundary image onto $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$4: $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$5 This is not described there as CMT in name, but it is a direct constrained transport formulation in which map admissibility is enforced through transport boundary conditions (Froese, 2011).

5. Variational discretizations and solver architectures

The algorithmic literature on CMT is correspondingly heterogeneous because the constraint geometry varies. A concise classification is given below.

Formulation	Representative mechanism	Representative paper
Boundary-constrained OT	Fixed-point updates of Neumann data for the transport boundary condition	(Froese, 2011)
Zero-pattern or capacity-constrained discrete OT	Sinkhorn-type scaling or double regularization with scalar root solves	(Corless et al., 2024, Wu et al., 2022)
Dynamic path-constrained OT	Augmented Lagrangian, PPXA, Uzawa, or Douglas–Rachford splitting	(Kerrache et al., 2022, Nishino et al., 10 Dec 2025, Wu et al., 2024)
Diffusion as transport	JKO-type finite-particle minimization with Wasserstein cost and entropy	(Pandolfi et al., 2023)

For discrete constrained OT with forbidden source–target pairs, the admissible transport matrix is restricted by a prescribed zero set $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$6: $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$7 A KL-regularized reformulation produces

$$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$8

where $$\frac{1}{\nu_m}\frac{dm}{d\tau} = \nabla\!\left[\nabla m-(n\nabla m-m\nabla n)\right],\qquad \frac{1}{\nu_n}\frac{dn}{d\tau} = \nabla\!\left[\nabla n-(m\nabla n-n\nabla m)\right],$$9 on $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$0. The optimizer retains a multiplicative form $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$1 on allowed entries, and iterative scaling algorithms of Sinkhorn–Knopp type converge to the unique optimum via alternating KL projections in the sense of Bregman (Corless et al., 2024).

For entrywise capacity constraints

$j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$2

double regularization replaces standard entropic OT by

$j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$3

The KKT conditions yield

$j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$4

so each alternating step reduces to solving scalar monotone equations $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$5, $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$6 by Newton’s method. The paper proves uniqueness of the regularized solution and convergence to the exact capacity-constrained optimum as $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$7 (Wu et al., 2022).

Dynamic constrained OT in the Benamou–Brenier setting is typically handled by saddle-point or proximal splitting methods. In the abstract constrained optimal transport framework, augmented Lagrangian splitting introduces auxiliary variables $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$8 and multipliers $j_{mn}=n\nabla m-m\nabla n=-j_{nm}.$9; convergence of the resulting algorithms is established under explicit parameter conditions such as

$\nu_n\ll \nu_m$0

Residuals $\nu_n\ll \nu_m$1, $\nu_n\ll \nu_m$2, and multiplier discrepancies converge to zero, and uniform convexity upgrades weak convergence to strong convergence (Kerrache et al., 2022).

In constrained unbalanced WFR, staggered-grid finite differences together with PPXA provide a modular numerical pipeline. The discrete problem is written as

$\nu_n\ll \nu_m$3

where $\nu_n\ll \nu_m$4 is the discrete WFR action, $\nu_n\ll \nu_m$5 encodes the continuity equation and boundary data, and each $\nu_n\ll \nu_m$6 is an affine box constraint on time slices. The new proximal ingredient is an orthogonal-box projection for affine inequality and equality constraints, while the prox operators for the WFR action and consistency constraint are inherited from unconstrained WFR numerics (Nishino et al., 10 Dec 2025).

For nonlinear controlled dynamics,

$\nu_n\ll \nu_m$7

the density evolves by

$\nu_n\ll \nu_m$8

and density constraints are enforced by indicator penalties $\nu_n\ll \nu_m$9. Input constraints are translated into linear constraints on momentum $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$00. Three solvers are proposed: a direct Uzawa-type method, an indirect Uzawa-type method based on a simpler Poisson equation, and a Douglas–Rachford splitting method for the convexified formulation. This extends CMT from free transport to affine-nonlinear controlled systems with both density and input constraints (Wu et al., 2024).

A separate algorithmic direction treats diffusion itself as a constrained transport problem. A JKO-type incremental functional

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$01

is discretized by finite-width Gaussian particles of fixed mass. The discrete transport penalty includes both position and width changes,

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$02

and the particle width is selected variationally, with asymptotic scaling

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$03

This is mass-conserving CMT in Wasserstein gradient-flow form rather than source-augmented OT (Pandolfi et al., 2023).

6. CMT in probability space and Boltzmann-generator learning

A recent machine-learning usage narrows CMT to a variational framework for learning Boltzmann generators by constructing a sequence of intermediate densities between a tractable base $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$04 and a target Boltzmann density $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$05. The motivation is that reverse-KL training,

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$06

is mode-seeking and prone to mode collapse, while fixed geometric annealing

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$07

can exhibit mass teleportation and requires schedule tuning. CMT replaces manual schedules with constrained variational updates in distribution space (Klitzing et al., 21 Oct 2025).

The trust-region formulation solves

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$08

while the entropy-constrained formulation imposes

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$09

The full CMT update combines both constraints. Each problem has a closed-form intermediate density. With both constraints active,

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$10

and the Lagrange multipliers are found by maximizing the associated dual (Klitzing et al., 21 Oct 2025).

The trust region is tied directly to local overlap and effective sample size: $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$11 Using $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$12, the paper derives the approximation

$$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$13

Thus the trust-region constraint stabilizes importance weights, while the entropy constraint prevents overly rapid concentration. In ablations, removing either constraint worsens coverage or stability; with both constraints, entropy decay is smoother and mode collapse is avoided (Klitzing et al., 21 Oct 2025).

The learned intermediate densities are approximated with normalizing flows via importance-weighted forward KL. Empirically, CMT is evaluated on alanine dipeptide, alanine tetrapeptide, alanine hexapeptide, and ELIL tetrapeptide. The abstract reports more than $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$14 higher effective sample size than prior variational methods. On ELIL tetrapeptide, the reported ESS is $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$15 for CMT, versus $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$16 for TA-BG and $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$17 for FAB; reverse KL drops to $$\partial_t \rho+\nabla\cdot m=0,\qquad \rho(0,\cdot)=\rho_0,\qquad \rho(1,\cdot)=\rho_1.$$18 and exhibits severe Ramachandran error. In this setting, CMT is literally a constrained mass-transport process in probability space, with constraints acting on successive KL displacement and entropy decay rather than on physical fluxes or densities (Klitzing et al., 21 Oct 2025).

Taken together, these developments show that the modern literature uses CMT to study constrained redistribution across several domains: particles moving through excluded-volume media, densities evolving under dynamic OT with path and source restrictions, flows limited by network or support constraints, and probability measures transported through controlled annealing paths. The unifying feature is not a single equation but the imposition of explicit admissibility structure on mass evolution, with the constraint set determining both the geometry of transport and the appropriate variational or numerical machinery.