Neural Operators with Exactly Linearizable Structure for Near-Integrable PDEs

Learned solution operators accumulate rollout error on systems whose evolution is exactly linear in the right coordinates. For near-integrable PDEs those coordinates exist in closed form: the inverse scattering transform (IST) diagonalizes the integrable flow. The Scattering-Coordinate Neural Operator (SCNO) routes the state through a differentiable nonlinear Fourier transform, applies the exact phase rotation, inverts by machine-precision Darboux dressing, and learns only the integrability-breaking residual. Zero-initialized, the pipeline is the analytic solution operator, verified to round-off; a learned transform cannot satisfy this identity. Soliton perturbation theory predicts that the residual, the learning target, grows as εz; we measure that law directly, and matched-capacity ablations show that the coordinates carry the advantage while three residual architectures inside them are interchangeable. We also measure where structure stops paying: across four fiber-physics perturbation channels the loss channel crosses over at ε* ∈ [0.020, 0.030] over baseline families, selection rules and budgets, beyond which tuned time-domain baselines win, and the same phase diagram re-emerges on damped Korteweg–de Vries through a different spectral problem and a different inverse. Removing the transmitter-side oracle and re-training from blindly recovered spectra changes the diagram by a median of 0%. Off the reflectionless manifold the error is set by the energy the coordinates exclude: a radiative fraction ρ costs ≈√ρ, flat in distance. The closest prior method, an FNO with learned scattering structure, is competent in distribution and diverges at twice its training horizon: learned approximate linearizations compound under extrapolation, exact ones cannot.