Learning in the Transverse Subspace: A Minimal Representation for Divergence-Free Operator Learning

Divergence-free vector fields are fundamental state variables in incompressible flows and many PDE systems. Redundant parameterizations, including Neural Conservation Law (NCL) potentials, map multiple auxiliary representations to the same physical divergence-free field. Our experiments show that this redundancy can reduce static representation-fitting error by enlarging the set of equivalent solutions. However, the resulting many-to-one mapping does not provide the unique function state required for operator learning, where each physical state must be evolved consistently to its future state. We introduce a minimal representation that encodes a real \(D\)-component divergence-free vector field on a \(D\)-dimensional domain as a real \((D-1)\)-component vector field on the same domain. The method exploits the transverse structure imposed by the divergence-free constraint in Fourier space and uses a Householder orthogonal transformation to construct the reduced coordinates directly. For periodic fields and closed impermeable fields, the transform is invertible, isometric, and angle-preserving. For open nonperiodic flows, we use Fourier extension to construct a compatible periodic field, while a minimum-energy rule selects a unique reduced representation. These coordinates provide a direct state space for divergence-free operator learning. Training data are first encoded into the \((D-1)\)-component representation, and the neural operator learns the temporal evolution entirely in this reduced space. At inference, the predicted reduced field is decoded directly into a physical \(D\)-component divergence-free field. The model therefore never predicts an ambient \(D\)-component field and does not require a post-hoc projection to remove longitudinal or divergent components. Experiments on static representation fitting and temporal prediction reveal a task-dependent trade-off. Redundant representations facilitate static optimization, whereas unique and invertible coordinates provide a well-defined state representation for learning temporal dynamics. In projection-based or other redundant formulations, unconstrained longitudinal or null directions can be discarded during projection or decoding and may introduce instability because they are not directly constrained by the physical loss. By eliminating these redundant directions from the learned state space, the proposed formulation achieves lower prediction error and greater robustness while maintaining the divergence-free constraint by construction.