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

<table> <tbody> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Fluid_Logo.html"><img src="docs/figures/examples/grids/Fluid_Logo.gif"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Wake_Flow.html"><img src="docs/figures/examples/grids/Wake_Flow.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Lid_Driven_Cavity.html"><img src="docs/figures/examples/grids/Lid_Driven_Cavity.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Taylor_Green.html"><img src="docs/figures/examples/grids/Taylor_Green.jpg"></a></td> </tr> <tr> <td align="center">Fluid logo</td> <td align="center">Wake flow</td> <td align="center">Lid-driven cavity</td> <td align="center">Taylor-Green</td> </tr> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Smoke_Plume.html"><img src="docs/figures/examples/grids/Smoke_Plume.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Variable_Boundaries.html"><img src="docs/figures/examples/grids/Variable_Boundaries.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Batched_Smoke.html"><img src="docs/figures/examples/grids/Batched_Smoke.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Moving_Obstacles.html"><img src="docs/figures/examples/grids/Moving_Obstacles.png"></a></td> </tr> <tr> <td align="center">Smoke plume</td> <td align="center">Variable boundaries</td> <td align="center">Parallel simulations</td> <td align="center">Moving obstacles</td> </tr> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Rotating_Bar.html"><img src="docs/figures/examples/grids/Rotating_Bar.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Multi_Grid_Fluid.html"><img src="docs/figures/examples/grids/Multi_Grid_Fluid.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Higher_order_Kolmogorov.html"><img src="docs/figures/examples/grids/Higher_Order_Kolmogorov.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Heat_Flow.html"><img src="docs/figures/examples/grids/Heat_Flow.png"></a></td> </tr> <tr> <td align="center">Rotating bar</td> <td align="center">Multi-grid fluid</td> <td align="center">Higher-order Kolmogorov</td> <td align="center">Heat flow</td> </tr> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Burgers.html"><img src="docs/figures/examples/grids/Burgers.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Reaction_Diffusion.html"><img src="docs/figures/examples/grids/Reaction_Diffusion.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Waves.html"><img src="docs/figures/examples/grids/Waves.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/grids/Julia_Set.html"><img src="docs/figures/examples/grids/Julia_Set.png"></a></td> </tr> <tr> <td align="center">Burgers' equation</td> <td align="center">Reaction-diffusion</td> <td align="center">Waves</td> <td align="center">Julia Set</td> </tr> </tbody> </table>

Mesh

<table> <tbody> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/mesh/FVM_BackStep.html"><img src="docs/figures/examples/mesh/FVM_BackStep.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/mesh/FVM_Heat.html"><img src="docs/figures/examples/mesh/FVM_Heat.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/mesh/Build_Mesh.html"><img src="docs/figures/examples/mesh/Build_Mesh.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/mesh/FVM_Cylinder_GMsh.html"><img src="docs/figures/examples/mesh/FVM_Cylinder_GMsh.png"></a></td> </tr> <tr> <td align="center">Backward facing step</td> <td align="center">Heat flow</td> <td align="center">Mesh construction</td> <td align="center">Wake flow</td> </tr> </tbody> </table>

Particles

<table> <tbody> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/SPH.html"><img src="docs/figures/examples/particles/SPH.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/FLIP.html"><img src="docs/figures/examples/particles/FLIP.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/Streamlines.html"><img src="docs/figures/examples/particles/Streamlines.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/Terrain.html"><img src="docs/figures/examples/particles/Terrain.jpg"></a></td> </tr> <tr> <td align="center">SPH</td> <td align="center">FLIP</td> <td align="center">Streamlines</td> <td align="center">Terrain</td> </tr> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/Gravity.html"><img src="docs/figures/examples/particles/Gravity.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/Billiards.html"><img src="docs/figures/examples/particles/Billiards.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/particles/Ropes.html"><img src="docs/figures/examples/particles/Ropes.png"></a></td> </tr> <tr> <td align="center">Gravity</td> <td align="center">Billiards</td> <td align="center">Ropes</td> </tr> </tbody> </table>

Optimization & Networks

<table> <tbody> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Gradient_Descent.html"><img src="docs/figures/examples/optim/Gradient_Descent.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Optimize_Throw.html"><img src="docs/figures/examples/optim/Optimize_Throw.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Learn_Throw.html"><img src="docs/figures/examples/optim/Learn_Throw.jpg"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/PIV.html"><img src="docs/figures/examples/optim/PIV.jpg"></a></td> </tr> <tr> <td align="center">Gradient Descent</td> <td align="center">Optimize throw</td> <td align="center">Learning to throw</td> <td align="center">PIV</td> </tr> <tr> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Close_Packing.html"><img src="docs/figures/examples/optim/Close_Packing.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Learn_Potential.html"><img src="docs/figures/examples/optim/Learn_Potential.png"></a></td> <td style="width: 25%;"><a href="https://tum-pbs.github.io/PhiFlow/examples/optim/Differentiable_Pressure.html"><img src="docs/figures/examples/optim/Differentiable_Pressure.jpg"></a></td> </tr> <tr> <td align="center">Close packing</td> <td align="center">Learning Φ(x,y)</td> <td align="center">Differentiable pressure</td> </tr> </tbody> </table>

Installation

Installation with pip on Python 3.6 and above:

$ pip install phiflow

Install PyTorch, [TensorFlow](https://www.te