AI4CFD: AI Methods for Computational Fluid Dynamics

Complete implementations of deep learning methods for solving Partial Differential Equations (PDEs), including tutorials, training code, and real-world applications.

🧠 Core Methods

1. Physics-Informed Neural Networks (PINNs)

Neural networks that encode physical laws directly into the loss function

Core Idea: Construct physics-constrained loss using automatic differentiation:

Loss = MSE(Boundary) + MSE(Initial) + MSE(PDE Residual)

• Advantages: Low data requirements, handles complex boundaries, suitable for inverse problems
• Location: PINNs/
• Tutorials: 8 Jupyter notebooks (English & Chinese), from basics to advanced
• Applications: Poisson, Heat, Navier-Stokes, Vlasov-Poisson systems

2. Deep Operator Networks (DeepONet)

Neural networks learning mappings between infinite-dimensional function spaces

Core Idea: Branch-Trunk architecture for learning operators G: u → G(u):

DeepONet(u)(y) = Σᵢ bᵢ(u) · tᵢ(y)
Branch network bᵢ: Encodes input function u
Trunk network tᵢ: Encodes output location y

• Advantages: Train once, fast inference (milliseconds), efficient for multi-query
• Location: DeepONet/
• Tutorials: Pure PyTorch implementation tutorial
• Applications: Heat equation, Burgers, Darcy flow, VP systems

3. Fourier Neural Operators (FNO)

Neural operators solving PDEs in frequency domain

Core Idea: Convolution in Fourier space for global information propagation:

v(x) = σ(W·u(x) + (K*u)(x))
where K*u computed in frequency domain: F⁻¹(R · F(u))

• Advantages: Resolution-invariant, excellent for periodic problems, high-resolution solving
• Location: FNO/
• Applications: Navier-Stokes, turbulence modeling, Darcy flow

4. Tensor Neural Networks (TNN)

Neural networks using tensor decomposition for high-dimensional PDEs

Core Idea: Decompose high-dimensional functions into products of low-dimensional functions:

u(x₁,...,xₐ) ≈ Σᵢ αᵢ · ∏ₖ φₖ⁽ⁱ⁾(xₖ)

• Advantages: Linear parameter growth (vs exponential), suitable for high-dimensional problems
• Location: TNN/
• Tutorials: Complete Jupyter tutorial and 5D examples
• Applications: 5D Poisson equation, high-dimensional PDE solving

5. Transformer-based Methods

Sequence models using attention mechanisms for PDE solving

Core Idea: Capture long-range dependencies in spatial/temporal domains via self-attention

• Advantages: Long-range dependency capture, flexible architecture design
• Location: Transformer/
• Applications: Time-series prediction, multi-physics coupling

📁 Project Structure

AI4CFD/
├── PINNs/              # Physics-Informed Neural Networks
│   ├── tutorial/       # 8 tutorials (English & Chinese)
│   ├── vp_system/      # Vlasov-Poisson system implementation
│   └── README.md
├── DeepONet/           # Deep Operator Networks
│   ├── tutorial/       # PyTorch implementation tutorial
│   ├── vp_system/      # VP operator learning
│   └── README.md
├── FNO/                # Fourier Neural Operators
│   └── README.md
├── TNN/                # Tensor Neural Networks
│   ├── tutorial/       # Complete tutorial
│   ├── train/dim5/     # 5D PDE solving example
│   └── README.md
├── Transformer/        # Transformer-based methods
│   └── README.md
├── VP_system/          # Vlasov-Poisson applications
│   └── TwoStreamInstability/  # Two-stream instability
└── utils/              # Shared utility functions

🚀 Quick Start

Installation

git clone https://github.com/Michael-Jackson666/AI4CFD.git
cd AI4CFD
pip install -r requirements.txt

Run Examples

PINNs Tutorial (Recommended for Beginners):

cd PINNs/tutorial
jupyter notebook tutorial_eng.ipynb  # English tutorial
# or
jupyter notebook tutorial_chinese.ipynb  # Chinese tutorial

TNN 5D Example:

cd TNN/train/dim5
python ex_5_1_dim5.py

DeepONet Tutorial:

cd DeepONet/tutorial
jupyter notebook operator_learning_torch.ipynb

📊 Method Comparison

Method	Training Data	Single Solve Speed	Parameter Query	Best For
PINNs	Low (physics-informed)	Seconds	Re-train needed	Complex boundaries, inverse problems, data scarcity
DeepONet	High (needs solutions)	Milliseconds	One forward pass	Multi-query, real-time prediction
FNO	High (needs solutions)	Milliseconds	One forward pass	Periodic problems, turbulence, high-resolution
TNN	Medium	Seconds	Re-train needed	High-dimensional problems (5D+)
Transformer	High (needs solutions)	Milliseconds	One forward pass	Time-series, long-range dependencies

🎯 Typical Applications

• Fluid Dynamics: Navier-Stokes equations, turbulence modeling, shape optimization
• Heat Transfer: Heat equation, convection-diffusion, multi-physics coupling
• Plasma Physics: Vlasov-Poisson systems, two-stream instability
• General PDEs: Poisson equation, Burgers equation, Darcy flow

🔧 Dependencies

• Python >= 3.8
• PyTorch >= 1.10
• NumPy, SciPy, Matplotlib
• Jupyter (optional, for tutorials)

📖 Documentation

Each method has detailed README documentation:

• PINNs/README.md: Complete PINNs guide and tutorial index
• DeepONet/README.md: Operator learning detailed explanation
• TNN/README.md: Tensor neural network theory and implementation
• TNN/train/dim5/README.md: 5D PDE solving example guide

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

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