Nonlocal Mean Field Schrödinger Bridge with Learned Interactions

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

Learning generative models in settings where the source and target distributions are only specified through unpaired samples is gaining in importance. Here, one frequently-used model are Schrödinger bridges (SB), which represent the most likely evolution between both endpoint distributions. To accelerate training, simulation-free SBs avoid the path simulation of the original SB models. However, learning simulation-free SBs requires paired data; a coupling of the source and target samples is obtained as the solution of the entropic optimal transport (OT) problem. As obtaining the optimal global coupling is infeasible in many practical cases, the entropic OT problem is iteratively solved on minibatches instead. Still, the repeated cost remains substantial and the locality can distort the global transport geometry. We propose quantized diffusion Schrödinger bridges (QDSB), which compute the endpoint coupling on anchor-quantized endpoint distributions and lift the resulting plan back to original data points through cell-wise sampling. We show that the regularized optimal coupling is stable w.r.t. anchor quantization, with an error controlled by the quality of the anchor approximation. In real-world experiments, QDSB matches the sample quality of existing baselines, requiring substantially less time. Code and data are available at github.com/mathefuchs/qdsb.

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics