Structure-Preserving Frozen Neural Operators

Coupled PDE solvers must enforce physical constraints while controlling optimization error and saddle-point conditioning. We introduce Structure-Preserving Frozen Neural Operators (SP-FNO), a frozen Fourier representation with coefficient-space constraints and gradient-free solves. A rank-revealing SVD yields an orthogonal projector for linear constraints. For mixed systems, we solve the velocity problem on the constraint kernel and recover pressure through a Schur complement governed by the retained inf-sup spectrum, independent of the constraint-block coupling scale. On boxes, analytic Gram assembly yields quadrature-free functional error majorants, valid with proven constants in exact arithmetic; our float64 evaluations do not enclose roundoff. The gradient-free core attains relative $L^2$ errors of $2.14\times 10^{-9}$, $1.50\times 10^{-9}$, and $1.90\times 10^{-4}$ on manufactured Stokes, Maxwell, and Navier–Stokes problems without frequency anchoring. For advection whose solution lies in the frozen span, it reaches $2.83\times 10^{-14}$ relative error in $1.19\times 10^{-2}$ seconds. A separate learned field-space hybrid achieves the best mean VRMSE on all five The Well datasets tested under the $4\to 1$ protocol, reaching $0.01668$ one-step VRMSE on active matter with a $17.34$M-parameter backbone and an eight-pass symmetry ensemble. Code: https://anonymous.4open.science/r/SP-FNO-C44A