Force Matching with Relativistic Constraints: A Physics-Inspired Approach to Stable and Efficient Generative Modeling

2025-02-12

AI Summary

This work addresses the instability and divergence in generative modeling caused by unbounded sampling velocities in continuous-time dynamics. We propose Force Matching (ForM), the first method to incorporate the special-relativistic Lorentz factor into the flow matching framework, explicitly constraining sample velocities to remain below the speed-of-light upper bound—thereby ensuring boundedness and stability of the entire sampling trajectory. We theoretically prove that this velocity constraint is rigorously preserved throughout the continuous-time dynamics. Empirically, ForM reduces Euclidean distance loss to 0.714 on the half-moons dataset, substantially outperforming baseline methods. Ablation studies confirm that the relativistic velocity constraint is critical for enhancing robustness. This work establishes a novel paradigm for designing generative dynamics guided by fundamental principles from relativistic physics.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

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 extit{half-moons} dataset, ForM significantly outperforms baseline methods, achieving the lowest Euclidean distance loss of extbf{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.

Problem

• Enhance generative model stability
• Incorporate relativistic velocity constraints
• Validate with empirical evaluations

Innovation

• Force Matching framework
• Lorentz factor constraint
• special relativity principles