Unbalanced Schrödinger Bridge Problem

• The Unbalanced Schrödinger Bridge Problem is a framework that interpolates between finite measures with varying mass using stochastic processes that incorporate birth and death events.
• It relies on regime-switching diffusions, coupled Fokker–Planck/Hamilton–Jacobi systems, and variational formulations to minimize relative entropy relative to a killed diffusion process.
• Recent advances include neural network solvers and iterative proportional fitting schemes, successfully applied to modeling cellular dynamics, epidemiological spread, and financial risk.

The unbalanced Schrödinger Bridge Problem (uSBP) is a rigorous framework for interpolating between finite (not necessarily equal-mass) measures via a stochastic process while permitting arbitrary mass variation through birth and death events. This generalization of classical Schrödinger bridge (SB) theory enables tractable modeling of physical, biological, and engineered systems where population size is not conserved, and is characterized by minimization of relative entropy on path space with respect to a reference process that includes killing (death) or growth (birth) mechanisms. Recent advances leverage regime-switching diffusions, dynamically coupled Fokker–Planck and Hamilton–Jacobi systems, and scalable neural algorithms to solve uSBP in both mean-field and interacting particle settings.

1. Mathematical Formulation and Regime-Switching Structure

The uSBP replaces the classical requirement of conservation of mass with flexible endpoint constraints that permit mass creation or annihilation. Let $\mu_0$ and $\mu_1$ be finite Borel measures on $\mathbb{R}^d$, which may have unequal mass. The reference process is typically a diffusion with killing, defined on the compactification $\hat{\mathbb{R}}^d = \mathbb{R}^d \cup \{\infty\}$, where the cemetery state $\infty$ records absorption/death events. The generator for such a process $(X_t)$ is

$$\hat K^0 f(x) = \begin{cases} \langle b(x), \nabla f(x) \rangle + \frac{1}{2} \Delta f(x) - k(x) \bigl(f(x) - f(\infty)\bigr), & x \in \mathbb{R}^d, \ 0, & x = \infty. \end{cases}$$

where $b(x)$ is a drift, $k(x) \geq 0$ the killing rate. The uSBP seeks a path measure $\mathbb{P}^{SB}$ minimizing

$$\mathbb{P}^{SB} = \arg\min_{\mathbb{P} : \mathbb{P}_0 = \mu_0, \, \mathbb{P}_T = \mu_1} KL(\mathbb{P} \mid \mathbb{P}^0),$$

where $\mathbb{P}^0$ is the law of the killed diffusion (Pariset et al., 2023).

In regime-switching diffusions, transition between active ("alive") and absorbing ("dead") regimes is governed by random killing times and possibly location- or path-dependent rates. The augmented path space includes a regime variable $\Lambda_t \in \{a, d\}$ and a jump map $\psi(t, x)$ for recording the death event (Zlotchevski et al., 15 Dec 2025). This structure admits a unified stochastic calculus representation for diverse unbalanced transport scenarios.

2. Dynamic, Variational, and Stochastic Control Formulations

The problem admits both dynamic and static representations. The dynamic (Benamou–Brenier-type) formulation is

$$\begin{aligned} \min_{(\rho, v, \kappa)} ~ & J(\rho, v, \kappa) = \int_0^T \int_{\mathbb{R}^d} \left\{\frac{1}{2} |v_t(x)|^2 \rho_t(x) + F(\kappa_t(x), \rho_t(x)) \right\} dx dt \ \text{subject to} ~ & \partial_t \rho_t + \nabla \cdot (\rho_t v_t) = \kappa_t \rho_t, \quad \rho_0 = \mu_0, \quad \rho_T = \mu_1, \end{aligned}$$

where $F$ is a convex penalty on the growth/killing rate $\kappa$, and $v_t(x)$ is the mass flow field (Zhang et al., 16 May 2025).

The stochastic control perspective (Zlotchevski et al., 15 Dec 2025, Zlotchevski et al., 8 Nov 2025) formulates uSBP as minimizing

$$\mathbb{E}\left[ \int_0^T \left( \frac{1}{2} \|u(t, X_t)\|^2 + V(t, X_t) [\xi(t, X_t)\log\xi(t, X_t) + 1 - \xi(t, X_t)] \right) dt \right]$$

over control drift $u(t, x)$ and jump (killing) rate modifier $\xi(t, x)$, subject to the SDE with killing rate $\xi(t, x)V(t, x)$ and target marginal constraints.

In regime-switching jump diffusion settings, the full bridge law is characterized by controls $(u^*, \theta^*, \xi^*)$ that are explicit functions of Schrödinger potentials (see Table 1).

Quantity	Formula	Reference
Optimal drift	$u^* = \sigma^\top \nabla_x \ln \varphi$	(Zlotchevski et al., 8 Nov 2025)
Optimal jump	$$\theta^*(t,x,z) = 1- \frac{\varphi(t,x+\gamma,1)}{\varphi(t,x,1)}$$	(Zlotchevski et al., 8 Nov 2025)
Optimal killing	$\xi^* = \frac{\varphi(t, x, 2)}{\varphi(t, x, 1)}$	(Zlotchevski et al., 8 Nov 2025)

The coupled forward–backward Kolmogorov (or Fokker–Planck) system encodes the time evolution of potentials and marginal densities under the bridge measure, with source (birth) and sink (killing) terms as appropriate.

3. Time Reversal, Birth–Death Duality, and Regime Extensions

A fundamental property is that time reversal of a diffusion with killing yields a process with birth terms. Under regularity, the time-reversed process $Y_t = X_{T-t}$ is Markovian with generator:

$$\hat B_t f(x) = \langle -b(x) + \nabla \ln p_{T-t}(x), \nabla f(x) \rangle + \frac{1}{2} \Delta f(x),$$

where the backward-time birth density is the forward killing rate weighted by the relative probability of survival, $k(y) p_{T-t}(y)/S_{T-t}$ (Pariset et al., 2023).

General regime-switching models further support:

• Arbitrary death maps $\psi$ (joint, marginal time/location, or only total loss constraints)
• Revivals or multiple absorbing states (by extending the state and jump structure)
• Integration of interactions and mean-field effects (Zhang et al., 16 May 2025, Zlotchevski et al., 15 Dec 2025)

The regime-switching perspective yields explicit Doob $h$-transform representations for the optimal bridge, applicable to both diffusions and jump diffusions.

4. Variational Algorithms and Deep Learning Solvers

Iterative proportional fitting (IPF) schemes extend to the unbalanced setting, alternating projections between prescribed initial and terminal (possibly sub-probability) marginals in the entropic space of path measures. Under suitable continuity and positivity, this yields convergence in KL to the unique uSBP solution (Pariset et al., 2023, Zlotchevski et al., 15 Dec 2025).

Neural parameterizations facilitate large-scale, high-dimensional applications. In (Pariset et al., 2023):

• Two families of score networks $f_\theta(t,x)$ and $\hat f_\vartheta(t,x)$ approximate potentials, while a scalar mass-loss parameter $\Psi$ enforces the mass constraint.
• Two algorithms are proposed:
  • UDSB-TD: introduces a temporal-difference (TD) loss for learning drifts and constants of integration, updating $\Psi$ via a closed-form IPF step.
  • UDSB-F ("Ferryman"): introduces a separate network $g_\zeta(t,x)$ for death rates and a "Ferryman" loss enforcing prescribed mass at one or more timepoints, enhancing stability.

For interacting particle systems and mean-field extensions, the CytoBridge algorithm (Zhang et al., 16 May 2025) uses neural nets to represent advection, growth rates, interactions, and score functions, solving the mean-field Fokker–Planck system via particle-based simulation and composite energy-reconstruction losses.

5. Theoretical Guarantees

Rigorous existence and uniqueness results for uSBP have been established under mild regularity—continuity and non-degeneracy of the drift and killing coefficients—using martingale problems and Föllmer–Girsanov duality (Pariset et al., 2023). In jump diffusion settings, sufficient regularity on the noise and jump structure yields classical ($C^{1,2}$) solutions to the coupled forward–backward system and guarantees the bridge law is itself a Markov process with explicit control drift and jump compensators (Zlotchevski et al., 8 Nov 2025).

The static and dynamic Schrödinger systems are well-posed for both general regime-switching and mean-field-interacting cases (Zlotchevski et al., 15 Dec 2025, Zhang et al., 16 May 2025). In hierarchical constraint families (killing time, space, full joint or only total mass known), KL divergence minimization obeys a lattice ordering reflecting the information content of the dead-regime constraints (Zlotchevski et al., 15 Dec 2025).

6. Representative Applications

The unbalanced SB framework has enabled novel analyses of systems with genuine mass variation:

• Single-cell drug response: UDSB-F accurately recovers population growth and shrinkage for transcriptomic cell states under drug application, outperforming balanced DSB models in both mean-matching and transport error metrics, and demonstrates biological plausibility in the localization of birth and death events (Pariset et al., 2023).
• COVID variant emergence: UDSB-F reconstructs trajectory of Delta variant fractions across Europe with higher fidelity than baselines, providing country-level intermediate-time inferences consistent with epidemiological spread data (Pariset et al., 2023).
• Cellular systems with interactions: CytoBridge recovers biologically meaningful state transitions, growth/shrinkage rates, and interaction patterns from sparse gene expression and scRNA-seq datasets, outperforming non-interacting and balanced baselines on distribution matching (Wasserstein $W_1$, total mass variation) (Zhang et al., 16 May 2025).

Additional applications include chemical reaction networks, modeling financial default/timing, credit-risk via absorbing regimes, and engineering systems with repair or failure dynamics (Zlotchevski et al., 15 Dec 2025, Zlotchevski et al., 8 Nov 2025).

7. Generalizations, Comparative Analysis, and Prospects

The regime-switching approach to uSBP accommodates diverse endpoint and intermediate constraints by mapping knowledge of killing time, location, or aggregated mass loss to abstract jump maps $\psi$ and hybrid path spaces (Zlotchevski et al., 15 Dec 2025). Comparative theorems guarantee monotonicity of the entropic cost with respect to constraint refinement, and the unified framework reconciles several prior unbalanced and excursion-bridge models.

Scalable numerical implementations include generalized Fortet–Sinkhorn iterations and score-based deep solvers for high-dimensional or interacting particle systems. Theoretical advances encompass mean-field propagation-of-chaos, strong convexity under entropic regularization, and error guarantees for randomized particle algorithms.

A plausible implication is that future research will extend these techniques to more general hybrid switching systems (multiple deaths/revivals), partially observed or noisy endpoint data, and domains where inference of survival or extinction dynamics is essential. The uSBP establishes a rigorous foundation for unbalanced dynamic optimal transport, bridging entropic control, stochastic analysis, and data-driven computational modeling.