Three-Body Scattering for Generative Modeling

An energy-distance field provides direct sample-level supervision for one-step generation, but sparse source interactions suffer from high noise. We propose Three-Body Scattering Modeling (TBSM), which learns a fake-to-real correction field via an online tracker that dynamically adapts to the evolving generated distribution. Under TBSM, each generated projectile is simultaneously attracted toward a real source and repelled from an independently generated source. At the exact energy-distance endpoint, the conditional expectation of this interaction coincides with the $2$-Wasserstein gradient-flow velocity of $\frac{1}{2}{D}_E^2(P_{\theta },Q)$. The online tracker approximates this expectation, while frozen-target regression transfers the estimated corrections into the generator, which is deployed standalone during inference. Theoretically, our analysis characterizes the regime where tracking provably reduces supervision error, establishing stationarity and convergence guarantees under explicit conditions. Computationally, each event conditions on a single real reference, requiring only $\mathcal{O}(B)$ sample interactions for a batch size of $B$. Leveraging frozen image features, TBSM trains one-step generators on ImageNet-256, achieving an $\mathrm{FID}=2.23$ with pixel-space PixelDiT-XL and an $\mathrm{FID}=1.63$ with latent-space DiT-XL at $\mathrm{NFE}=1$. Finally, we provide a unified design map relating this learning paradigm to diffusion supervision, drift dynamics, and GAN-like objectives.