[Paper Review] Fourier Neural Operator for Parametric Partial Differential Equations

Zongyi Li, Nikola Kovachki+5 more|arXiv (Cornell University)|Oct 18, 2020

Model Reduction and Neural Networks32 references1,078 citations

TL;DR

The paper introduces the Fourier Neural Operator (FNO), a mesh-invariant neural operator that learns mappings between function spaces by parameterizing kernels in Fourier space, achieving fast, zero-shot super-resolution on PDEs like Burgers, Darcy, and Navier-Stokes.

ABSTRACT

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.

Motivation & Objective

• Develop a neural operator framework that maps from function spaces to function spaces for parametric PDEs.
• Achieve mesh-invariance and zero-shot super-resolution by parameterizing kernels in Fourier space.
• Demonstrate speedups and accuracy gains over traditional solvers and prior learning-based methods across multiple PDEs.
• Show applicability to time-dependent and time-independent PDEs and discuss data requirements and limitations.

Proposed method

• Define an iterative neural operator architecture where v_{t+1}(x) = σ(W v_t(x) + (K(a; φ) v_t)(x)).
• Replace the kernel integral operator with a Fourier-space convolution operator to enable efficient computation via FFT.
• Parameterize the Fourier kernel κ_φ through its Fourier transform R_φ and truncate to k_max modes for tractable learning.
• Use four Fourier integral operator layers with ReLU activations and batch normalization to form the FNO.
• Demonstrate discretization-invariance and zero-shot super-resolution by training on low-resolution data and evaluating on higher-resolution grids.

Experimental results

Research questions

• RQ1 Can a neural operator learn the solution operator for parametric PDEs in an input-function to output-function setting?
• RQ2 Does parameterizing the kernel in Fourier space yield mesh-invariant, fast, and accurate operators across different PDEs and resolutions?
• RQ3 Can the Fourier neural operator achieve zero-shot super-resolution and outperform existing neural operators and baselines on Burgers’, Darcy, and Navier–Stokes equations?
• RQ4 What are the data requirements and trade-offs for training FNOs for complex PDEs, and how do they perform in Bayesian inverse problems?

Key findings

• The Fourier neural operator learns resolution-invariant solution operators for Burgers’, Darcy, and Navier–Stokes in turbulent regimes.
• FNO achieves zero-shot super-resolution and can be evaluated at higher resolutions than seen during training.
• On fixed resolutions, FNO outperforms benchmarks by roughly 30% for Burgers’, ~60% for Darcy, and ~30% for Navier–Stokes in relative error.
• On a 256×256 grid, FNO inference time is 0.005 s versus 2.2 s for a pseudo-spectral solver, with no accuracy loss in downstream tasks like Bayesian inference.
• FNO-3D (space-time convolution) often provides best performance when data are sufficient; FNO–2D variants still outperform several baselines when data are limited.
• The model remains effective with non-periodic boundaries and can recover high-frequency content through activations between layers.