Fourier Neural Operator

In this project, we introduce a powerful new class of neural networks — the Fourier Neural Operator (FNO) — designed to efficiently learn operators arising from partial differential equations (PDEs).

By parameterizing the integral kernel directly in Fourier space, we develop an expressive and scalable architecture that outperforms existing methods in both accuracy and speed.

Our model has been extensively tested on:

• Burgers' Equation
• Darcy Flow
• Navier-Stokes Equation (including turbulent regimes)

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🚀 Highlights

• Efficient Kernel Learning: Directly learning in Fourier space drastically reduces computational complexity.
• Fast and Scalable: Achieves up to three orders of magnitude faster inference compared to traditional PDE solvers.
• State-of-the-art Performance: Outperforms existing neural network-based methods on a wide range of benchmarks.
• Simple and Modular Code: All scripts are standalone, clean, and easy to adapt for different applications.

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📚 Requirements

• PyTorch

Install the required library using:

pip install torch

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📁 Project Structure

Each script in this repository is independent and directly runnable.

• main.py — Training loop and evaluation scripts.
• utilities.py — Dataset generation, loading, and preprocessing utilities.
• models/ — Model architecture files.
• data/ — Datasets for different PDE problems.

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📦 Datasets

We provide datasets for the Burgers equation and Darcy flow.
Data generation scripts are available in utilities.py.

• PDE Datasets

Download and place them inside the data/ directory.

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🧠 Pretrained Models

Evaluate the pre-trained models easily using the provided scripts like _eval.py or _super_resolution.py.

• Pretrained Models

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⚙️ How to Run

Train the model:

python main.py

Evaluate the model:

python eval.py

Super-resolution tasks:

python super_resolution.py

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📈 Results

Problem	FNO Performance	Traditional Solvers
Burgers' Equation	3x faster	Slower
Darcy Flow	100x faster	Much slower
Navier-Stokes Equation	State-of-the-art	Poor generalization

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✨ Conclusion

The Fourier Neural Operator provides an efficient, scalable, and highly accurate method for learning PDE mappings.
By working in Fourier space, we achieve faster training, better generalization, and superior performance over classical methods and traditional neural networks.

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📩 Contact

For any questions or collaborations, feel free to reach out!