FNO (Fourier Neural Operator)¶

FNO learns an operator mapping using Fourier-domain global mixing, enabling efficient modeling of long-range interactions.

Reference Paper: Li et al., ICLR 2021.

@inproceedings{
Li2020FourierNO,
title={Fourier neural operator for parametric partial differential equations},
author={Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
booktitle={The Ninth International Conference on Learning Representations},
year={2021}
}

RealPDEBench interface¶

• Input: x with shape [B, T_in, H, W, C_in]
• Output: predictions with shape [B, T_out, H, W, C_out]

Implementation notes (RealPDEBench)¶

• The baseline appends a normalized coordinate grid (t, x, y) before lifting.
• It uses a vectorized reshape for multi-step outputs, which assumes \(T_\text{out}\) is a multiple of \(T_\text{in}\).
• Training uses an MSE-style objective on predicted fields.

FNO specific YAML config¶

This baseline is enabled by model_name: "fno" in the training YAML.

Config files¶

• Cylinder: realpdebench/configs/cylinder/fno.yaml
• Controlled Cylinder: realpdebench/configs/controlled_cylinder/fno.yaml
• FSI: realpdebench/configs/fsi/fno.yaml
• Foil: realpdebench/configs/foil/fno.yaml
• Combustion: realpdebench/configs/combustion/fno.yaml

Model-specific keys¶

These keys are consumed by realpdebench.model.load_model.load_model() and realpdebench.model.fno.FNO3d.

• modes1 (int): Number of kept Fourier modes along the first FFT dimension (upper bound \( \lfloor N/2 \rfloor + 1 \)).
• modes2 (int): Number of kept Fourier modes along the second FFT dimension.
• modes3 (int): Number of kept Fourier modes along the third FFT dimension.
• n_layers (int): Number of stacked Fourier blocks (each mixes a spectral convolution + a pointwise \(1\times1\times1\) conv).
• width (int): Channel width after lifting (latent feature dimension used throughout the network).