# From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

arXiv: https://arxiv.org/abs/2607.00320 · HTML: https://mlsanity.com/paper/2607.00320/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 59 references that are in the map.
- First posted on arXiv 1 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [Operator learning meets inverse problems: A probabilistic perspective](https://mlsanity.com/paper/2508.20207/) (2025)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators](https://mlsanity.com/paper/2408.09004/) (2024)
- [Convergence Rates for Learning Pseudo-Differential Operators](https://mlsanity.com/paper/2601.04473/) (2026)
- [Continuous Data Assimilation with Learned Surrogate Dynamics](https://mlsanity.com/paper/2606.00480/) (2026)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)

## Abstract (arXiv)

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-T solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.