Variational Auto Encoder with CFD

A simulation of wake behind cylinder. dimensionality reduction by variational auto encoder

Introduction

1. Numerical simulation of wake behind a cylinder

This is a simple simulation of wake behind a cylinder. The simulation is done using Lattice Boltzmann Method. ( see cylinder.cpp )

The simulation is done for five different Reynolds numbers($Re = 5, 40, 60, 100, 200$).

2. Dimensionality reduction by Variational Auto Encoder

The simulation data is then used to train an variational auto encoder to reduce the dimensionality of the data to 32-sized latent space (vae.py). This took about 10 minutes on single RTX 3090 GPU.

3. Neural network to predict time integral of latent space

We then defined a neural network to predict time integral step() function on the latent space. Neural network takes 32-sized latent vector z and Reynolds number $Re$ as input and predicts the next latent vector z'. (stepper.py)

We will see that the neural network is able to predict the next latent vector with untrained Reynolds number.

TODO List

• Test LSTM for latent stepper - ( first 10 steps must be calculated by real numeric simulation )
• [] Test Transformer for latent stepper

How to run

creating simulated training data by LBM

$ mkdir build
cmake ..
make
./CylinderLBM

This will create re5.dat, re40.dat, re60.dat, re100.dat and re200.dat in current directory.

training VAE

python ../vae.py

This will train VAE and save the model vae.pt in current directory.

training LatentStepper

python ../stepper.py

This will train LatentStepper and save the model stepper.pt in current directory.

plotting latent-simulation result

python ../plotter.py ReynoldsNumber

This performs the simulation on latent space and saves the result in plots$Re/ directory.

making gif & mp4

sh ../png2gif.sh ReynoldsNumber

sh ../png2mp4.sh ReynoldsNumber

Results

Loss of VAE

Snapshots of compressed result

Snapshot of compressed result of $Re = 200$. One horizontal line represents compressed snapshot of specific time step, and the y-axis represents the time step.

Loss of LatentStepper (Training loss)

Loss of LatentStepper with untrained Reynolds number ($Re=150$)

For each iteration, the $L_2$ error is calculated as:

$ \sum_{x=0}^{512} \sum_{y=0}^{256} |predicted(x,y) - simulated(x,y)|^2 / (256*512) $

and the $L_{inf}$ error is calculated as:

$ \max_{pixel\in image} |predicted(pixel) - simulated(pixel)| $

Result of Untrained Reynolds number, reconstructed from VAE

$Re = 20$

result/out20.gif

$Re = 150$

result/out150.gif