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

The recent introduction of the non-conservative generalized Schrödinger bridge (NCGSB) marks a significant advancement in modeling stochastic processes, as it allows for energy variation over time, overcoming the limitations imposed by traditional energy-conservation assumptions. This reformulation is grounded in contact Hamiltonian mechanics and captures more complex dynamics, making it applicable to a broader range of real-world phenomena. The NCGSB's implementation through contact Wasserstein geodesics (CWG) leverages a ResNet architecture, facilitating efficient computation with near-linear complexity. This non-iterative approach not only enhances performance but also supports guided generation through task-specific distance metrics. The framework has been validated across various applications, including manifold navigation and molecular dynamics predictions, demonstrating its versatility and practical benefits in advancing computational methods in stochastic modeling.