Tensor Field Models Enhance Conditional Generative AI

Generative AI models are constantly evolving, seeking more structured and efficient ways to create complex data. This research introduces Tensor Field Models (TFMs), a novel mathematical framework designed to enhance conditional generation. TFMs are defined as realization-level structures where a learned operator maps a product of admissible component-section families to a specific family of time-dependent tangent sections on a generative state manifold. A key aspect of TFMs is their ability to encode analytic and dynamical restrictions through the selection of admissible families, rather than imposing them rigidly. The framework includes structured refinements like constructed, component-separable, and Tensor Bundle TFMs. In the conditional realizations explored, a structured condition is mapped component-wise to a reusable collection of representations. The architectures evaluated maintain distinct component representations, combining them only through the Field Operator to produce the generated Vector Field. All learned models are trained using Flow Matching. Experiments demonstrate that TFMs can significantly improve performance, and the use of reusable condition representations enables amortized sampling, which in turn accelerates the generation process.