Use Fourier transform to learn operators in differential equations.

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:

• (GKN) Neural Operator: Graph Kernel Network for Partial Differential Equations
• (MGKN) Multipole Graph Neural Operator for Parametric Partial Differential Equations

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}
}