Generative Modeling via Nonlinear Schr\"odinger Bridges and Branching Diffusions

arXiv:2610.29476v1 Announce Type: new 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 Schr\"odinger 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 Schr\"odinger bridge problem through a coupled system of PDEs, yielding a model we call SBNL (Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger bridges.