Φ_Flow is an open-source simulation toolkit built for optimization and machine learning applications. It is written mostly in Python and can be used with NumPy, PyTorch, Jax or TensorFlow. The close integration with these machine learning frameworks allows it to leverage their automatic differentiation functionality, making it easy to build end-to-end differentiable functions involving both learning models and physics simulations.

Examples

Grids

Fluid logo	Wake flow	Lid-driven cavity	Taylor-Green
Smoke plume	Variable boundaries	Parallel simulations	Moving obstacles
Rotating bar	Multi-grid fluid	Higher-order Kolmogorov	Heat flow
Burgers' equation	Reaction-diffusion	Waves	Julia Set

Mesh

Backward facing step	Heat flow	Mesh construction	Wake flow

Particles

SPH	FLIP	Streamlines	Terrain
Gravity	Billiards	Ropes

Optimization & Networks

Gradient Descent	Optimize throw	Learning to throw	PIV
Close packing	Learning Φ(x,y)	Differentiable pressure

Installation

Installation with pip on Python 3.6 and above:

$ pip install phiflow

Install PyTorch, TensorFlow or Jax in addition to Φ_Flow to enable machine learning capabilities and GPU execution. To enable the web UI, also install Dash. For optimal GPU performance, you may compile the custom CUDA operators, see the detailed installation instructions.

You can verify your installation by running

$ python3 -c "import phi; phi.verify()"

This will check for compatible PyTorch, Jax and TensorFlow installations as well.

Features

• Tight integration with PyTorch, Jax and TensorFlow for straightforward neural network training with fully differentiable simulations that can run on the GPU.
• Built-in PDE operations with focus on fluid phenomena, allowing for concise formulation of simulations.
• Flexible, easy-to-use web interface featuring live visualizations and interactive controls that can affect simulations or network training on the fly.
• Object-oriented, vectorized design for expressive code, ease of use, flexibility and extensibility.
• Reusable simulation code, independent of backend and dimensionality, i.e. the exact same code can run a 2D fluid sim using NumPy and a 3D fluid sim on the GPU using TensorFlow or PyTorch.
• High-level linear equation solver with automated sparse matrix generation.

📖 Documentation and Tutorials

Documentation Overview   •   ▶ YouTube Tutorials   •   API   •   Demos   •   Playground

Φ-Flow builds on the tensor functionality from Φ_ML. To understand how Φ_Flow works, check named and typed dimensions first.

Getting started

• Installation instructions
• Tensors
• Fluids
• Cookbook

Physics

• Grid-based fluids
• Higher-order schemes

Fields

• Overview
• Staggered grids
• I/O & scene format

Geometry

• Overview
• Signed distance fields
• Heightmaps

Tensors

• ▶️ Introduction Video
• [Introduction Notebook](Math_In