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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

Korean

우리는 앞으로 뒤로 부분적인 프레임 차원균형 (PDEs)에 기초한 새로운 트랙티브 모델링 프레임 워크를 제안합니다. 이러한 균형은 중간 필드 게임 (MFG) 이론에서 자연적으로 발생하고 비선형 Schrödinger 브리지 시스템과 밀접하게 관련되어 있습니다. 우리의 주요 공헌은 생성형 모델링의 설정 내에서이 연결을 재해석하고 실용적인 구현을위한 효율적인 숫자 방법론을 개발하는 것입니다. 특히, 우리는 SBNL (Schrödinger Bridge NonLinear)라고 부르는 모델을 제공하는 PDEs의 부합된 시스템을 통해 고전적인 Schrödinger 브리지 문제의 비선형 확장을 추출하고 특징화합니다. 우리는 분포 분포와 동일한 반선형 PDEs의 확률적 표현을

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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