Tensor Field Models

Abstract

This paper introduces Tensor Field Models (TFMs), realization-level Mathematical Structures in which a learned Operator maps a product of admissible component-section families to a prescribed family of time-dependent tangent sections on a Generative State Manifold. Analytic and dynamical restrictions are encoded through the choice of admissible families rather than imposed by the root definition. Constructed, component-separable, and Tensor Bundle TFMs provide structured refinements of this common object. In the conditional realizations considered here, a structured condition $c=(c_1,\ldots,c_n)$ is mapped componentwise to a reusable collection $\mathbf H_c=(H_{c_1}^{(1)},\ldots,H_{c_n}^{(n)})$. In the architectures evaluated here, the component representations remain distinct and are combined only by the Field Operator to produce the generated Vector Field. All learned models are trained using Flow Matching. Experiments show that TFMs can improve performance and that amortized sampling enabled by reusable condition representations can accelerate generation.