Title: Contact Wasserstein Geodesics for Non-Conservative Schrödinger Bridges
Primary area: generative models
Keywords: Schrödinger Bridge, Generative Models, Hamiltonian, Contact Hamiltonian, Differential Geometry, Wasserstein metric

Abstract: 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.

TL;DR: We introduce a non-conservative reformulation of the Schrödinger Bridge problem, along with a ResNet parameterization that can efficiently learn its solution.