Haar Paraproduct Neural Operator

Physical-field prediction involves interactions between regional means and internal variations. Bounding individual coupling coefficients, however, need not control their accumulated effect across a hierarchy. We introduce the Haar Paraproduct Neural Operator (HPNO), which learns these interactions through two input-conditioned bounded functions, called symbols. Their Haar coefficients define a mean-to-detail paraproduct and a detail-to-mean adjoint paraproduct on a weighted binary hierarchy, while a bounded Haar multiplier reweights existing components. Regional orthogonality converts pointwise symbol bounds into joint control of squared transfer coefficients within every subtree. This yields a weighted $L^2$ bound for each fixed-input kernel action, independent of hierarchy size and depth. We also characterize both directional responses across arbitrary tree cuts and express their exact norms through regional symbol variation. HPNO achieves the lowest reported relative ${\ell }_2$ errors among the compared methods on all six PDE benchmarks spanning regular grids, structured meshes, and point clouds, with competitive results on two unstructured computational fluid dynamics benchmarks. Ablations on Darcy and Elasticity support bidirectional coupling and bounded-symbol construction. Paired Darcy tests further show lower regional response errors than HNO and LinearNO under permeability rearrangements that preserve regional input means.