arXiv cs.AI· Mathias Grau, Pierre Henry-Labord\`ere·· 13 小时前AI 评分24
基于非线性薛定谔桥与分支扩散的生成建模框架 SBNL
Generative Modeling via Nonlinear Schr\"odinger Bridges and Branching Diffusions
AI 导读
研究者提出 SBNL（Schrödinger Bridge NonLinear），通过耦合 PDE 系统将经典薛定谔桥问题推广到非线性情形，并基于分支扩散与半线性 PDE 的概率表示设计求解算法。该方法可高度并行、扩展至高维，适用于广泛的非线性交互泛函，并在 MNIST 与 FFHQ 人脸数据集的高维图像生成任务上完成验证。同一框架还可嵌入基于分数的生成建模、扩散模型与可处理的高斯薛定谔桥。
正文 · 原文
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.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML) |
| Cite as: | arXiv:2610.29476 [cs.LG] |
| (or arXiv:2610.29476v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.29476 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Mathias Grau [view email]
[v1]
Wed, 21 Oct 2026 09:30:53 UTC (12,382 KB)
来源：arXiv cs.AI · arxiv.org