Fourier Neural Operators get polynomial sample complexity guarantees

A new paper from researchers Chandramoorthy, Sanz-Alonso, and Waniorek provides rigorous theoretical foundations for Fourier Neural Operators (FNOs) in learning solution operators of dissipative evolution equations. The authors prove that FNOs can efficiently approximate and learn these operators whenever stable and accurate spectral discretizations exist. They introduce classes of evolution operators defined through spectral methods and derive approximation bounds alongside polynomial sample complexity guarantees. For equations with polynomial nonlinearities, learning rates primarily depend on input space smoothness and physical domain dimension. The results hold uniformly across broad families of dissipative PDEs, including Navier-Stokes, Allen-Cahn, and Cahn-Hilliard equations.

For equations with non-polynomial smooth nonlinearities, the analysis shows polynomial sample complexity still holds, but rates additionally depend on the smoothness of nonlinear terms and dissipation strength. This 66-page preprint bridges classical spectral approximation theory with modern operator learning, explaining when FNOs can efficiently learn nonlinear evolution operators. The findings are significant for scientific machine learning, potentially reducing the data requirements for training neural operators in computational physics and engineering applications where PDE solvers are essential.