Paradifferential Neural Operators: Better Call Bony For Non-Linear PDEs

Neural Operators have emerged as powerful architectures for learning mappings between infinite-dimensional function spaces. While predominantly deployed as PDE surrogates, their broader promise in data-driven AI for Science requires models capable of rigorously resolving complex non-linear dynamics. To address this, we introduce ParaNO, a Fourier-based neural operator grounded in paradifferential calculus. Crucially, ParaNO leverages Bony's decomposition to efficiently learn the cross-scale frequency interactions caused by non-linearities. Alongside our method, we introduce The Non-Linear Six (The NL6), a curated benchmark designed to systematically isolate distinct classes of algebraic non-linearities without geometric confounding. ParaNO improves over SotA baselines by 23% on Navier–Stokes and reduces errors by factors of 2x to 4.6x on the NL6. Our code, benchmark, and network weights will be made available upon publication.