Fourier Neural Operators (FNOs) are a class of machine learning models designed for solving partial differential equations (PDEs) and learning mappings between function spaces. They extend neural operators by incorporating the Fourier transform to capture global dependencies efficiently. Unlike traditional deep learning models, which work with pointwise approximations, FNOs operate in the frequency domain, making them highly efficient for problems involving complex spatial patterns.

Key Concepts of Fourier Neural Operators

1. Neural Operators
   • Unlike standard neural networks that map finite-dimensional vectors to vectors, neural operators learn mappings between infinite-dimensional function spaces.
   • Useful for solving PDEs, turbulence modeling, and climate forecasting.
2. Fourier Transform for Global Feature Extraction
   • Instead of operating in the spatial domain, FNOs apply a Fourier transform to the input function.
   • The model filters out high-frequency noise and focuses on essential frequency components.
   • Enables efficient learning of global dependencies.
3. Fourier Layer (Spectral Convolution)
   • A key innovation of FNOs is the Fourier layer, where convolutions are performed in the Fourier domain.
   • This allows capturing long-range dependencies with fewer parameters compared to traditional convolutional neural networks (CNNs).
4. Architecture of Fourier Neural Operators
   • The input function is lifted to a higher-dimensional space via a neural network.
   • The Fourier transform is applied to obtain frequency components.
   • A spectral convolution (multiplication with a learned weight matrix) is performed in the Fourier domain.
   • The inverse Fourier transform is applied to return to the spatial domain.
   • A final neural network maps the result back to the output function space.

Advantages of FNOs

✅ Efficiency – Fewer parameters and faster training than conventional deep learning approaches.

✅ Scalability – Works well with high-dimensional PDEs without requiring dense spatial discretization.

✅ Long-range dependencies – Captures global patterns, unlike CNNs which rely on local convolutions.

✅ Generalization – Can learn solutions for a family of PDEs, not just a single instance.

Applications

• Fluid Dynamics (Navier-Stokes equations, turbulence modeling)
• Weather and Climate Prediction
• Computational Physics
• Biomedical Imaging
• Material Science and Engineering

🧠Implementation of FNO (PyTorch Example)

https://gist.github.com/viadean/41f02702b4f237f7038eab8cbdfefa78