This paper introduces Force Matching (ForM), a novel framework for generative modeling that represents an initial exploration into leveraging special relativistic mechanics to enhance the stability of the sampling process. By incorporating the Lorentz factor, ForM imposes a velocity constraint, ensuring that sample velocities remain bounded within a constant limit. This constraint serves as a fundamental mechanism for stabilizing the generative dynamics, leading to a more robust and controlled sampling process. We provide a rigorous theoretical analysis demonstrating that the velocity constraint is preserved throughout the sampling procedure within the ForM framework. To validate the effectiveness of our approach, we conduct extensive empirical evaluations. On the \textit{half-moons} dataset, ForM significantly outperforms baseline methods, achieving the lowest Euclidean distance loss of \textbf{0.714}, in contrast to vanilla first-order flow matching (5.853) and first- and second-order flow matching (5.793). Additionally, we perform an ablation study to further investigate the impact of our velocity constraint, reaffirming the superiority of ForM in stabilizing the generative process. The theoretical guarantees and empirical results underscore the potential of integrating special relativity principles into generative modeling. Our findings suggest that ForM provides a promising pathway toward achieving stable, efficient, and flexible generative processes. This work lays the foundation for future advancements in high-dimensional generative modeling, opening new avenues for the application of physical principles in machine learning.

翻译：本文提出力匹配（ForM）这一生成建模新框架，首次探索利用狭义相对论力学增强采样过程的稳定性。通过引入洛伦兹因子，ForM施加速度约束，确保样本速度始终有界于恒定范围内。该约束作为稳定生成动力学的基本机制，使采样过程更具鲁棒性和可控性。我们提供了严格的理论分析，证明在ForM框架下速度约束能在整个采样过程中保持。为验证方法的有效性，我们开展了广泛的实证评估。在\textit{half-moons}数据集上，ForM显著优于基线方法，取得了最低的欧几里得距离损失（\textbf{0.714}），而原始一阶流匹配（5.853）与一阶及二阶流匹配（5.793）的损失值均较高。此外，我们通过消融实验进一步探究速度约束的影响，再次证实ForM在稳定生成过程方面的优越性。理论保证与实证结果共同表明，将狭义相对论原理融入生成建模具有巨大潜力。我们的研究发现，ForM为实现稳定、高效且灵活的生成过程提供了可行路径。本工作为高维生成建模的未来发展奠定基础，为物理原理在机器学习中的应用开辟了新途径。