Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions

Mathias Grau; Pierre Henry-Labordère

cs.LGOct 21, 2026

Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions

Authors: Mathias Grau, Pierre Henry-Labordère

Organizations: †Qube Research & Technologies, Paris, France.

Abstract

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 ).

\widehat{\varphi}_{k}(x)\!\leftarrow\!\tfrac{1}{M}\!\sum_{m=1}^{M}\!\mathcal{W}^{m}\!\prod_{i=1}^{N_{T}^{m,\mathrm{w}}}\!\varphi_{\theta}(X_{T,x}^{m,i})\!\prod_{j=1}^{\bar{N}_{T}^{m,\mathrm{b}}}\!\bar{\varphi}_{\eta}(\bar{X}_{T,x}^{m,j})

Algorithm 2 Sampling from semilinear SB

Figure 2. Temporal evolution of $\mathrm{Law}(X_{t})$ from Moons $\to$ Swiss (source $\mu_{0}$ on the left, target $\mu_{T}$ on the right).

Panel of Figure 7 — 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 $\hat{u}_{M}(0,x)$ , in the branching case and the heat kernel, highlighting the impact of the nonlinearity.

Panel of Figure 9 — Figure 9. First three contributions in the perturbative expansion (Gross-Pitaevskii cubic branching). Dashed lines represent white particles ( $\varphi$ ), solid lines represent black particles ( $\bar{\varphi}$ ). Weights with $a_{1}=p_{1}=1$ : order $c^{0}$ , no branching, i.e. classical Schrödinger bridge / linear heat kernel; order $c^{1}$ with one branching at $s$ (white offspring starting at $s$ , black offspring at $T-s$ ); order $c^{2}$ with two branchings at $s_{1},s_{2}$ : offspring of the same type as the parent start at $s_{i}$ , while offspring of the other type start at $T-s_{i}$ . The horizontal time axes (all at the same vertical level) show time from $0$ to $T$ , with dashed vertical lines indicating the splitting times.

Panel of Figure 10 — Figure 10. Classical Schrödinger bridges in $\mathbb{R}^{2}$ .

Figure 11. Extended view of the temporal evolution of $\mathrm{Law}(X_{t})$ (Moons $\to$ Swiss).

Figure 12. Terminal samples on Gaussian $\to$ mixture of $8$ Gaussians for each method.

Panel of Figure 13 — Figure 13. Terminal samples per method on the Gaussian $\to$ mixture of $8$ Gaussians benchmark.

Panel of Figure 14 — Figure 14. Per-method comparison on the Gaussian $\to$ mixture of $8$ Gaussians benchmark (lower is better; mean $\pm$ std over $5$ bootstraps).

Figure 15. Per-mode coverage of the $8$ target components. The dashed line marks the uniform reference $1/8$ .

Panel of Figure 16 — Figure 16. Gated maze: SBNL transport and estimated density evolution $m(t,\cdot)=\varphi(t,\cdot)\bar{\varphi}(T-t,\cdot)$ .

Panel of Figure 17 — Figure 17. S-curve: SBNL transport and estimated density evolution $m(t,\cdot)=\varphi(t,\cdot)\bar{\varphi}(T-t,\cdot)$ .

Panel of Figure 18 — Figure 18. Valley: SBNL transport and estimated density evolution $m(t,\cdot)=\varphi(t,\cdot)\bar{\varphi}(T-t,\cdot)$ .

Panel of Figure 19 — Figure 19. Spiral: SBNL transport and estimated density evolution $m(t,\cdot)=\varphi(t,\cdot)\bar{\varphi}(T-t,\cdot)$ .

Panel of Figure 20 — Figure 20. Three Gaussians: SBNL transport and estimated density evolution $m(t,\cdot)=\varphi(t,\cdot)\bar{\varphi}(T-t,\cdot)$ .

Figure 21. Generated vs. true samples in $\mathbb{R}^{50}$ (PCA projection).

Figure 22. Temporal evolution in $\mathbb{R}^{50}$ (PCA projection).

Panel of Figure 23 — Figure 23. Evolution of the first three moments (mean, standard deviation, skewness) of the generated samples over training epochs, for different latent dimensions.

Panel of Figure 24 — Figure 24. MNIST latent bridge ( $d{=}32$ ): latent PCA visualization and decoded samples.

Figure 25. FFHQ young-to-old: decoded trajectories during sampling.

Figure 26. FFHQ young-to-old comparison in ALAE latent space, extended panel.

Panel of Figure 27 — Figure 27. Convergence of $\mathrm{Law}(X_{T})$ with training (Moon curves $\to$ Swiss).

Panel of Figure 28 — Figure 28. Evolution of potential $\varphi$ (Moon curves $\to$ Swiss).

Panel of Figure 29 — Figure 29. Evolution of potential $\bar{\varphi}$ (Moon curves $\to$ Swiss).

Panel of Figure 30 — Figure 30. Flow evolution for $\lambda\in\{10^{5},10^{4},5\!\times\!10^{3},10^{3},10^{2}\}$ .

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Foundations of Schrödinger Bridges for Generative Modeling

Sophia Tang

Department of Computer and Information Science University of Pennsylvania

Schrödinger Bridges Generative Modeling

Open paper AskPDF arXiv

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.

Abstract

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