Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions
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.
Related
- Multi-type branching diffusions with small mutation rates
- Extinction and extinguishment properties for a nonlinear predator-prey branching model
- Global well-posedness for the derivative nonlinear Schrödinger equation on the circle
- Power-Type Nonlinear Schrödinger Equations with the Logarithmic Laplacian
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
BibTeX
@misc{grau2026generative,
title = {{Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions}},
author = {Mathias Grau and Pierre Henry-Labordère},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29476},
eprint = {2610.29476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29476},
}
RIS
TY - UNPB
TI - Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions
AU - Mathias Grau
AU - Pierre Henry-Labordère
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29476
UR - https://arxiv.org/abs/2610.29476
AB - 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.
ER -