The accurate prediction of geometric state evolution in complex systems is critical for advancing scientific domains such as quantum chemistry and material modeling. Traditional experimental and computational methods face challenges in terms of environmental constraints and computational demands, while current deep learning approaches still fall short in terms of precision and generality. In this work, we introduce the Geometric Diffusion Bridge (GDB), a novel generative modeling framework that accurately bridges initial and target geometric states. GDB leverages a probabilistic approach to evolve geometric state distributions, employing an equivariant diffusion bridge derived by a modified version of Doob's $h$-transform for connecting geometric states. This tailored diffusion process is anchored by initial and target geometric states as fixed endpoints and governed by equivariant transition kernels. Moreover, trajectory data can be seamlessly leveraged in our GDB framework by using a chain of equivariant diffusion bridges, providing a more detailed and accurate characterization of evolution dynamics. Theoretically, we conduct a thorough examination to confirm our framework's ability to preserve joint distributions of geometric states and capability to completely model the underlying dynamics inducing trajectory distributions with negligible error. Experimental evaluations across various real-world scenarios show that GDB surpasses existing state-of-the-art approaches, opening up a new pathway for accurately bridging geometric states and tackling crucial scientific challenges with improved accuracy and applicability.

翻译：复杂系统中几何状态演化的精确预测对于推进量子化学和材料建模等科学领域至关重要。传统的实验和计算方法面临环境约束和计算需求方面的挑战，而当前的深度学习方法在精度和泛化性方面仍存在不足。本文提出几何扩散桥（Geometric Diffusion Bridge, GDB），这是一种新颖的生成建模框架，能够精确连接初始与目标几何状态。GDB采用概率方法演化几何状态分布，利用通过改进版Doob $h$-变换推导出的等变扩散桥来连接几何状态。该定制化扩散过程以初始和目标几何状态作为固定端点，并由等变转移核控制。此外，通过使用等变扩散桥链，轨迹数据可在GDB框架中被无缝利用，从而提供更精细准确的演化动力学表征。理论上，我们通过全面分析证实了该框架能够保持几何状态的联合分布，并能以可忽略误差完全建模诱导轨迹分布的基础动力学。在多种实际场景中的实验评估表明，GDB超越了现有最先进方法，为精确连接几何状态以及以更高精度和适用性解决关键科学挑战开辟了新途径。