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

Yifei Sun

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 field. Our experiments show that this redundancy can reduce static representation-fitting error by enlarging the set of equivalent solutions, but the resulting many-to-one mapping does not provide a unique state for operator learning. 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 field on the same domain. Exploiting the transverse structure imposed by incompressibility in Fourier space, we use a Householder orthogonal transformation to construct the reduced coordinates directly. For periodic and closed impermeable fields, the transform is invertible, isometric, and angle-preserving. For open nonperiodic flows, Fourier extension constructs a compatible periodic field, and a minimum-energy rule selects a unique reduced representation. Neural operators then learn temporal evolution entirely in this reduced space. At…