We propose a new generative modeling framework grounded in forward-backward partial differential equations (PDEs). These equations arise naturally in mean-field game (MFG) theory, and are closely related to nonlinear Schrödinger systems. Our main contribution is to reinterpret this connection within the setting of generative modeling and to develop an efficient numerical methodology for its practical implementation. Specifically, we derive and characterize a nonlinear extension of the classical Schrödinger bridge problem through a coupled system of PDEs, yielding a model we call SBNL (Schrödinger Bridge NonLinear). To compute its solution, we design a simulation algorithm based on branching diffusions and the probabilistic representation of semilinear PDEs. The resulting method is embarrassingly parallel, scalable to high dimensions, and applicable to a broad class of nonlinear interaction functionals. We validate the approach through numerical experiments, including recovery of the classical Schrödinger bridge, nonlinear extensions with convex penalties (Gross-Pitaevskii) as well as potential interaction terms, and high-dimensional generative tasks such as image synthesis on the MNIST and FFHQ human face dataset. Furthermore, the same framework embeds score-based generative modeling, diffusion models, and tractable gaussian Schrödinger bridges.
Figures & tables
Figure 1. Branching paths on an 8-Gaussian terminal potential, illustrating Gross-Pitaevskii-type branching (Appendix M ).
Figure 2. Temporal evolution of Law(Xt) from Moons → Swiss (source μ0 on the left, target μT on the right).
Figure 4Figure 5Figure 6
Figure 6. Temporal evolution of the latent variables and their corresponding decoded representations.
Figure 7. FFHQ young-to-old in ALAE latent space. Lower FID is better.
Appendix figures & tables23 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8. Comparison of the analytic solution u(0,x) with the Monte Carlo estimate u^M(0,x) , in the branching case and the heat kernel, highlighting the impact of the nonlinearity.
Figure 9. First three contributions in the perturbative expansion (Gross-Pitaevskii cubic branching). Dashed lines represent white particles ( φ ), solid lines represent black particles ( φˉ ). Weights with a1=p1=1 : order c0 , no branching, i.e. classical Schrödinger bridge / linear heat kernel; order c1 with one branching at s (white offspring starting at s , black offspring at T−s ); order c2 with two branchings at s1,s2 : offspring of the same type as the parent start at si , while offspring of the other type start at T−si . The horizontal time axes (all at the same vertical level) show time from 0 to T , with dashed vertical lines indicating the splitting times.
Figure 10. Classical Schrödinger bridges in R2 .
Figure 11. Extended view of the temporal evolution of Law(Xt) (Moons → Swiss).
Figure 12. Terminal samples on Gaussian → mixture of 8 Gaussians for each method.
Figure 13. Terminal samples per method on the Gaussian → mixture of 8 Gaussians benchmark.
Figure 14. Per-method comparison on the Gaussian → mixture of 8 Gaussians benchmark (lower is better; mean ± std over 5 bootstraps).
Figure 15. Per-mode coverage of the 8 target components. The dashed line marks the uniform reference 1/8 .
Figure 16. Gated maze: SBNL transport and estimated density evolution m(t,⋅)=φ(t,⋅)φˉ(T−t,⋅) .
Figure 17. S-curve: SBNL transport and estimated density evolution m(t,⋅)=φ(t,⋅)φˉ(T−t,⋅) .
Figure 18. Valley: SBNL transport and estimated density evolution m(t,⋅)=φ(t,⋅)φˉ(T−t,⋅) .
Figure 19. Spiral: SBNL transport and estimated density evolution m(t,⋅)=φ(t,⋅)φˉ(T−t,⋅) .
Figure 20. Three Gaussians: SBNL transport and estimated density evolution m(t,⋅)=φ(t,⋅)φˉ(T−t,⋅) .
Figure 21. Generated vs. true samples in R50 (PCA projection).
Figure 22. Temporal evolution in R50 (PCA projection).
Figure 23. Evolution of the first three moments (mean, standard deviation, skewness) of the generated samples over training epochs, for different latent dimensions.
At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schrödinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schrödinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schrödinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.
Sophia Tang
Department of Computer and Information Science University of Pennsylvania
Diffusion models often yield highly curved trajectories and noisy score targets due to an uninformative, memoryless forward process that induces independent data-noise coupling. We propose Adjoint Schrödinger Bridge Matching (ASBM), a generative modeling framework that recovers optimal trajectories in high dimensions via two stages. First, we view the Schrödinger Bridge (SB) forward dynamic as a coupling construction problem and learn it through a data-to-energy sampling perspective that transports data to an energy-defined prior. Then, we learn the backward generative dynamic with a simple matching loss supervised by the induced optimal coupling. By operating in a non-memoryless regime, ASBM produces significantly straighter and more efficient sampling paths. Compared to prior works, ASBM scales to high-dimensional data with notably improved stability and efficiency. Extensive experiments on image generation show that ASBM improves fidelity with fewer sampling steps. We further showcase the effectiveness of our optimal trajectory via distillation to a one-step generator.
Jeongwoo Shin, Jinhwan Sul, Joonseok Lee +2
Seoul National University, Seoul, Korea · Georgia Institute of Technology, Atlanta, GA, USA · Sungkyunkwan University, Seoul, Korea
Recent advances in generative modeling have enabled the efficient computation of Schrödinger bridges (SB) in high-dimensional settings by leveraging partially simulation-free training methods inspired by flow matching. However, these have not covered SBs with reflecting dynamics, a useful model choice with built-in guarantees that generated samples stay in the data domain. Existing alternatives for reflected SBs instead rely on more complex training based on forward--backward SDE theory, requiring expensive higher-order derivatives and sampling entire paths during training. In this article, we introduce a partially simulation-free framework that allows reflected SBs to be trained similarly to flow matching, using a new sampling method and regression target. We demonstrate our results by coupling pairs of well-known high-dimensional image datasets. Using reflected dynamics incurs negligible additional wall-clock time during both training and inference while maintaining or slightly improving generative performance.
Marcus Häggbom, Viktor Nilsson, Pierre Nyquist +1
SEB Group and Department of Mathematics KTH Royal Institute of Technology Stockholm, Sweden · Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg and Department of Mathematics KTH Royal Institute of Technology