标题： 深度广义薛定谔桥：从图像生成到求解平均场博弈 显示英文标题

标题： Deep Generalized Schrödinger Bridges: From Image Generation to Solving Mean-Field Games

摘要： 广义薛定谔桥（GSBs）是一个基本的数学框架，用于根据最小作用原理分析最可能的粒子演化，包括动能和势能。 在量子力学和最优传输的理论领域中，它们有着广泛的应用，本文则从算法的角度出发，旨在提高其实际应用性。 我们的观察是，由GSBs界定最优结构的运输问题在各种科学领域中普遍存在，例如机器学习中的生成建模、随机控制中的平均场博弈等。 因此，探索GSBs的数学建模与现代算法特征之间的内在联系，是一个重要但尚未开发的研究方向。 在本文中，我们将GSBs重新解释为概率模型，并证明通过一种称为非线性费曼-卡茨引理的精妙数学工具，GSBs的最优结构自然地涌现出丰富的算法概念，如似然性、变分间隙和时间差分。 由此产生的计算框架，由深度学习和神经网络驱动，在完全连续的状态空间（即无网格）中运行，并满足分布约束，这使其区别于依赖空间离散化或约束放松的先前数值求解器。 我们在生成建模和平均场博弈中展示了我们方法的有效性，突显了数学建模、随机过程、控制和机器学习交叉领域的变革性应用。 显示英文摘要

摘要： Generalized Schr\"odinger Bridges (GSBs) are a fundamental mathematical framework used to analyze the most likely particle evolution based on the principle of least action including kinetic and potential energy. In parallel to their well-established presence in the theoretical realms of quantum mechanics and optimal transport, this paper focuses on an algorithmic perspective, aiming to enhance practical usage. Our motivated observation is that transportation problems with the optimality structures delineated by GSBs are pervasive across various scientific domains, such as generative modeling in machine learning, mean-field games in stochastic control, and more. Exploring the intrinsic connection between the mathematical modeling of GSBs and the modern algorithmic characterization therefore presents a crucial, yet untapped, avenue. In this paper, we reinterpret GSBs as probabilistic models and demonstrate that, with a delicate mathematical tool known as the nonlinear Feynman-Kac lemma, rich algorithmic concepts, such as likelihoods, variational gaps, and temporal differences, emerge naturally from the optimality structures of GSBs. The resulting computational framework, driven by deep learning and neural networks, operates in a fully continuous state space (i.e., mesh-free) and satisfies distribution constraints, setting it apart from prior numerical solvers relying on spatial discretization or constraint relaxation. We demonstrate the efficacy of our method in generative modeling and mean-field games, highlighting its transformative applications at the intersection of mathematical modeling, stochastic process, control, and machine learning.