Contact Wasserstein Geodesics for Non-Conservative Schr""odinger Bridges

2025-11-10

AI Summary

Existing stochastic bridge methods rely on energy conservation assumptions, limiting their ability to model dynamics with time-varying energy. This work proposes the Non-Conservative Generalized Schrödinger Bridge (NCGSB), the first framework to integrate contact Hamiltonian mechanics into probabilistic bridge modeling—explicitly capturing energy evolution. By parameterizing the Wasserstein manifold, NCGSB reduces the infinite-dimensional bridge problem to a finite-dimensional geodesic computation, enabling non-iterative, near-linear-time solution. The method unifies contact geometry, Wasserstein geodesics, and ResNet-inspired architecture, and introduces a task-adaptive distance metric. Evaluated on manifold navigation, molecular dynamics prediction, and image generation, NCGSB achieves significant improvements in intermediate-state fidelity and sample quality, demonstrating both theoretical rigor and practical efficacy.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

The Schr""odinger 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 Schr""odinger 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.

Problem

• Overcoming energy conservation limitations in Schrödinger bridge modeling
• Enabling energy-varying stochastic processes for real-world dynamics
• Providing efficient non-iterative computation for bridge problems

Innovation

• Non-conservative generalized Schrödinger bridge for energy-varying processes
• Parameterizing Wasserstein manifold for tractable geodesic computation
• Contact Wasserstein geodesic with near-linear complexity ResNet implementation