# Fourier Neural Operator for Parametric Partial Differential Equations

arXiv: https://arxiv.org/abs/2010.08895 · HTML: https://mlsanity.com/paper/2010.08895/

## Key facts

- Cited by 2,908 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2020.

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## Abstract (arXiv)

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.