The Schrödinger Bridge provides a principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-energy phenomena. To overcome this, we introduce the non-conservative generalized Schrödinger bridge (NCGSB), a novel, energy-varying reformulation based on contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in a finite-dimensional space. Unlike computationally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via a ResNet architecture and relies on a non-iterative solver with near-linear complexity. Furthermore, CWG supports guided generation by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility.

翻译：薛定谔桥为分布间的随机过程建模提供了一个原理性框架；然而，现有方法受限于能量守恒假设，这限制了桥的形状，使其无法模拟能量变化的现象。为克服此限制，我们引入了非保守广义薛定谔桥，这是一种基于接触哈密顿力学、具有能量变化特性的新颖重构。通过允许能量随时间变化，NCGSB 为现实世界的随机过程提供了一个更广泛的类别，能够捕捉更丰富、更忠实的时间演化动力学。通过对沃瑟斯坦流形进行参数化，我们将桥问题提升到一个有限维空间中可处理的测地线计算。与计算昂贵的迭代解法不同，我们的接触沃瑟斯坦测地线自然地通过 ResNet 架构实现，并依赖于一个具有近线性复杂度的非迭代求解器。此外，CWG 通过调制任务特定的距离度量来支持引导生成。我们在流形导航、分子动力学预测和图像生成等任务上验证了我们的框架，证明了其实用优势和多功能性。