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

TL;DR

Force Matching (ForM) introduces a physics-inspired, relativistic framework for generative modeling that constrains sample velocity via the Lorentz factor to stabilize sampling. By formulating a second-order sampling dynamics and proving a strict velocity bound $\|\dot{x}_t\|_2 < c$, ForM achieves robust, controlled trajectories and integrates with TrigFlow for flexible interpolation. Empirically, ForM outperforms standard Flow Matching baselines on Onedot, Halfmoons, and Spiral datasets, attaining lower Euclidean distance losses (e.g., Onedot $0.509$, Halfmoons $0.714$, Spiral $0.124$) and demonstrating the value of relativistic forces in guiding distribution transport. The work highlights theoretical guarantees, ablations validating component contributions, and future directions toward high-dimensional, adaptive, and hybrid physics-informed generative models.

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.

Figures (4)

• Figure 1: Left: Onedot Dataset. The objective is to train the ForM model to learn a transport trajectory from distribution $\pi_0$ (blue) to distribution $\pi_1$ (pink). Right: The transportation trajectory generated by the ForM model.
• Figure 2: Left: Halfmoons Dataset. The objective is to train ForM to learn a transport trajectory from distribution $\pi_0$ (blue) to distribution $\pi_1$ (pink). Right: The transportation trajectory generated by the ForM model.
• Figure 3: Left: Spiral Dataset. The objective is to train the ForM model to learn a transport trajectory from distribution $\pi_0$ (blue) to distribution $\pi_1$ (pink). Right: The transportation trajectory generated by the ForM model.
• Figure 4: Left: Flow matching lcb+22 using only the first-order term. Middle: An improved method that incorporates both first- and second-order terms. Right: Our proposed ForM model applied to the Onedot, Halfmoons, and Spiral datasets. Note that the first-order method (O1) fails to capture the target distribution, and although the second-order method (O2) exhibits slight improvement, it still does not adequately model the target distribution. In contrast, by leveraging the Lorentz force to guide the trajectory evolution, the ForM model significantly enhances the accuracy of the target distribution, as further evidenced by the quantitative results in Table \ref{['tab:euclidean_distance_complex_datasets_new']}.

Theorems & Definitions (18)

• Definition 3.1: Lorentz Factor
• Definition 3.2: Proper Time
• Definition 3.3: Relativistic Force
• Lemma 3.4: Equivalent Form of Relativistic Force, informal version of Lemma \ref{['lem:equiv_relativistic_force:formal']}
• Definition 4.1: Force Matching Objective
• Theorem 4.2: Sampling ODE, informal version of Theorem \ref{['thm:ode_form:formal']}
• Theorem 4.3: Speed Limit, informal version of Theorem \ref{['thm:vel:formal']}
• Theorem 4.4: ForM with TrigFlow, informal version of Theorem \ref{['thm:form_trig:formal']}
• Theorem A.1: Sampling ODE, formal version of Theorem \ref{['thm:ode_form:informal']}
• proof