A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge

Asmaa Eldesoukey; Yongxin Chen; Abhishek Halder

math.OCApr 8, 2026

A Generalized Sinkhorn Algorithm for Mean-Field Schrödinger Bridge

Authors: Asmaa Eldesoukey, Yongxin Chen, Abhishek Halder

Organizations: Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA · School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

Figures & tables


	$\displaystyle\partial_{t}\varphi_{t}^{\{i+1\}}+\sigma^{2}\langle b_{t}^{(k)}$	$\displaystyle,\nabla\varphi_{t}^{\{i+1\}}\rangle+\frac{\sigma^{2}}{2}\Delta\varphi_{t}^{\{i+1\}}$
		$\displaystyle=\sigma^{2}Q_{t}(\varphi_{t}^{j},p_{t}^{(k)})\varphi_{t}^{\{i+1\}},$	(11a)
with $\varphi_{t=1}^{\{i+1\}}(x)=\varphi_{1}^{\{i\}}(x)$ .

Algorithm 1 Mean-Field Sinkhorn

Schrödinger Bridges

Schrödinger bridges are stochastic transports that match prescribed distributions while staying close to a reference process.

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Nonlocal Mean Field Schrödinger Bridge with Learned Interactions

Daisuke Inoue, Dante Kalise, Mathieu Laurière

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom. · Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, Shanghai 200126, People’s Republic of China.

Neural Surrogate Modeling Schrödinger Bridges Mean-Field Theory

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The Schrödinger Bridge Problem connects an initial distribution to a terminal one along a minimum-energy stochastic process. Its mean-field extension, the Mean-Field Schrödinger Bridge, governs interacting populations whose dynamics and costs depend on the collective distribution. When these interactions are nonlocal, their direct evaluation scales quadratically with the population size, making large ensembles intractable within FBSDE-based solvers. We replace these terms with neural surrogates in state and time, trained on empirical interaction values along sampled trajectories and embedded in a four-stage alternating scheme that updates the forward and backward potentials and the surrogates in turn, while preserving forward--backward consistency and the prescribed endpoint marginals. We derive Grönwall-type stability bounds quantifying how surrogate errors propagate to the generated trajectories under a small-gain condition. On crowd-navigation and high-dimensional opinion-dynamics benchmarks, the surrogates reproduce the trajectories obtained with exact evaluation at reduced training cost. The advantage is most significant when the interaction is a nonlinear functional of the measure, such as the normalized bounded-confidence drift, for which random-batch subsampling is biased and unstable whereas the learned surrogate remains accurate.

Abstract

Schrödinger Bridges

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