Force Matching with Relativistic Constraints: A Physics-Inspired Approach to Stable and Efficient Generative Modeling (2502.08150v1)

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 \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.

• The paper introduces Force Matching (ForM), a generative modeling framework that integrates special relativistic mechanics to impose robust stability during sampling.
• ForM applies relativistic principles by leveraging the Lorentz factor to set velocity constraints, using a second-order ODE for more stable and expressive generative paths.
• Empirical tests show ForM outperforms baselines, achieving superior stability with theoretical velocity bounds and a significantly lower Euclidean distance loss on datasets like half-moons.

The paper "Force Matching with Relativistic Constraints: A Physics-Inspired Approach to Stable and Efficient Generative Modeling" introduces Force Matching (ForM), a distinct framework in generative modeling that integrates special relativistic mechanics to impose robust stability within the sampling process. This framework applies relativistic principles by leveraging the Lorentz factor to establish velocity constraints on generated samples, consequently stabilizing and controlling the dynamics within the generative model effectively.

The theoretical contribution of the paper is centered on providing a rigorous analysis ensuring that these velocity constraints are maintained consistently throughout the sampling process. The framework adopts a second-order ordinary differential equation (ODE) to solve the generative path based on the defined relativistic force, offering more expressive and stable sampling trajectories compared to conventional first-order flow models. Specifically, this ODE is presented as: $$\ddot{x}_t = \frac{1}{m^{\text{lab}} \gamma_t}\left(f_t^{\text{local}} - \frac{\langle v_t^{\text{lab}}, f_t^{\text{local}} \rangle}{c^2} v_t^{\text{lab}}\right)$$ Here, $\gamma_t$ is the Lorentz factor, $f_t^{\text{local}}$ is the relativistic force, $v_t^{\text{lab}}$ the velocity, $c$ the speed of light, $m^{\text{lab}}$ the mass, and $t$ represents time.

The ForM framework allows for a velocity constraint that remains below $c$, the speed of light, at all times. This theoretical constraint ensures sample velocities do not reach unstable magnitudes, thus avoiding the pitfalls of numerical instability frequently encountered in flow- or diffusion-based models through excessive velocity magnitudes. Moreover, it establishes theoretical guarantees on velocity bounds, highlighting the framework's robustness.

Empirical validation on datasets such as the half-moons dataset demonstrated ForM's superiority over baseline methods, including vanilla first-order and first- and second-order flow matching, achieving the lowest Euclidean distance loss of 0.714 compared to their 5.853 and 5.793 losses, respectively. An ablation paper further confirmed the stability benefits imparted by the velocity constraint mechanism of ForM.

The paper's findings underscore the potential of embedding principles from special relativistic mechanics into generative models. This integration suggests a promising trajectory towards stable, efficient, and scalable generative processes, with ForM laying the groundwork for future exploration of physics-inspired mechanisms within machine learning.