Generative Anchored Fields: Controlled Data Generation via Emergent Velocity Fields and Transport Algebra

We present Generative Anchored Fields (GAF), a generative model that learns independent endpoint predictors, $J$ (noise) and $K$ (data), from any point on a linear bridge. Unlike existing approaches that use a single trajectory or score predictor, GAF is trained to recover the bridge endpoints directly via coordinate learning. The velocity field $v=K-J$ emerges from their time-conditioned disagreement. This factorization enables \textit{Transport Algebra}: algebraic operations on multiple $J/K$ heads for compositional control. With class-specific $K_n$ heads, GAF defines directed transport maps between a shared base noise distribution and multiple data domains, allowing controllable interpolation, multi-class composition, and semantic editing. This is achieved either directly on the predicted data coordinates ($K$) using Iterative Endpoint Refinement (IER), a novel sampler that achieves high-quality generation in $5-8$ steps, or on the emergent velocity field ($v$). We achieve strong sample quality (FID 7.51 on ImageNet $256\times256$ and $7.27$ on CelebA-HQ $256\times 256$, without classifier-free guidance) while treating compositional generation as an architectural primitive. Code available at https://github.com/IDLabMedia/GAF.