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Generative Modeling via Nonlinear Schrödinger Bridges and Branching Diffusions

Mathias Grau, Pierre Henry‐Labordère · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.29476 · cited by 0 works (OpenAlex) · open-access PDF

Spanish

Proponemos un nuevo marco de modelado generativo basado en ecuaciones diferenciales parciales de marco (PDEs) hacia adelante y hacia atrás. Estas ecuaciones surgen naturalmente en la teoría del juego de campo medio (MFG) y están estrechamente relacionadas con los sistemas no lineales de Schrödinger. Nuestra contribución principal es reinterpretar esta conexión dentro de la configuración de modelado generativo y desarrollar una metodología numérica eficiente para su implementación práctica. En concreto, derivamos y caracterizamos una extensión no lineal del problema de puente de Schrödinger clásico a través de un sistema de PDEs acoplados, produciendo un modelo que llamamos SBNL (Schrödinger Bridge NonLinear). Para calcular su solución, desarrollamos un algoritmo de simulación basado en las difusiones de rama y la representación

Machine translation (Meta M2M100); the English original follows.

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.

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Topics: Nonlinear system, Generative grammar, Mathematics, Probabilistic logic, Computer science, Applied mathematics

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