Fourier Neural Operator

This repository contains the code for the paper:

(FNO) Fourier Neural Operator for Parametric Partial Differential Equations

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 the Navier-Stokes equation (including the turbulent regime). Our Fourier neural operator shows state-of-the-art performance compared to existing neural network methodologies and it is up to three orders of magnitude faster compared to traditional PDE solvers.

It follows from the previous works:

Requirements

We have updated the files to support PyTorch 1.8.0. Pytorch 1.8.0 starts to support complex numbers and it has a new implementation of FFT. As a result the code is about 30% faster.
Previous version for PyTorch 1.6.0 is avaiable at FNO-torch.1.6.

Files

The code is in the form of simple scripts. Each script shall be stand-alone and directly runnable.

fourier_1d.py is the Fourier Neural Operator for 1D problem such as the (time-independent) Burgers equation discussed in Section 5.1 in the paper.
fourier_2d.py is the Fourier Neural Operator for 2D problem such as the Darcy Flow discussed in Section 5.2 in the paper.
fourier_2d_time.py is the Fourier Neural Operator for 2D problem such as the Navier-Stokes equation discussed in Section 5.3 in the paper, which uses a recurrent structure to propagates in time.
fourier_3d.py is the Fourier Neural Operator for 3D problem such as the Navier-Stokes equation discussed in Section 5.3 in the paper, which takes the 2D spatial + 1D temporal equation directly as a 3D problem
The lowrank methods are similar. These scripts are the Lowrank neural operators for the corresponding settings.
data_generation are the conventional solvers we used to generate the datasets for the Burgers equation, Darcy flow, and Navier-Stokes equation.

Datasets

We provide the Burgers equation, Darcy flow, and Navier-Stokes equation datasets we used in the paper. The data generation configuration can be found in the paper.

PDE datasets

The datasets are given in the form of matlab file. They can be loaded with the scripts provided in utilities.py. Each data file is loaded as a tensor. The first index is the samples; the rest of indices are the discretization. For example,

Burgers_R10.mat contains the dataset for the Burgers equation. It is of the shape [1000, 8192], meaning it has 1000 training samples on a grid of 8192.
NavierStokes_V1e-3_N5000_T50.mat contains the dataset for the 2D Navier-Stokes equation. It is of the shape [5000, 64, 64, 50], meaning it has 5000 training samples on a grid of (64, 64) with 50 time steps.

We also provide the data generation scripts at data_generation.

Models

Here are the pre-trained models. It can be evaluated using eval.py or super_resolution.py.

models

Citations

@misc{li2020fourier,
      title={Fourier Neural Operator for Parametric Partial Differential Equations}, 
      author={Zongyi Li and Nikola Kovachki and Kamyar Azizzadenesheli and Burigede Liu and Kaushik Bhattacharya and Andrew Stuart and Anima Anandkumar},
      year={2020},
      eprint={2010.08895},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@misc{li2020neural,
      title={Neural Operator: Graph Kernel Network for Partial Differential Equations}, 
      author={Zongyi Li and Nikola Kovachki and Kamyar Azizzadenesheli and Burigede Liu and Kaushik Bhattacharya and Andrew Stuart and Anima Anandkumar},
      year={2020},
      eprint={2003.03485},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}