NeuralClosure

Tutorials for learning neural closure models for various discretized PDEs, including

• the viscous Burgers equations in 1D
• the incompressible Navier-Stokes equations in 2D

The tutorials are provided as Julia scripts. Markdown and Jupyter notebook versions are auto-generated from the Julia source files using Literate.jl.

Burgers	Navier-Stokes
Source file	Source file
Markdown	Markdown
Notebook	Notebook

See the script make.jl for converting the script to markdown or notebook.

Example simulations

Running the tutorials

It is recommended to start with the Burgers tutorial. The tutorial includes some exercises, but you do not need to know Julia to do them. Setting up the tutorial environment may take a few minutes. With the default parameters, the tutorial should run in about a minute on a modern laptop (including training).

Option 1: Running locally on your machine

To run the tutorials locally, you need to install Julia on your machine. An editor such as VSCode can also be useful. Proposed workflow:

1. Install Julia from one of the two options:

   • the official Juliaup version manager. This is the preferred way, as you will get notified about updates to Julia. It requires typing a line into your command line. See the Juliaup README for installation instructions.
   • the official downloads page (select the binary for your platform)

   Do not install Julia from your platform's package manager (e.g. apt on Ubuntu), as these versions may have flaws).

2. Install VSCode

3. Install the Julia extension for VSCode. This gives syntax highlighting, code execution with Shift + Enter etc.

4. Clone the repository in VSCode:

   1. Create a new window: Ctrl/Cmd + Shift + N

   2. Open the Source control side panel with Ctrl/Cmd + Shift + G

   3. Click the button Clone Repository. If you do not see this button, there should be an instruction there instead to first install Git.

      

   4. Paste the following URL:

       https://github.com/agdestein/NeuralClosure/
      

   5. Select a location for where you want the folder

   6. Open the cloned repository

5. Activate the project: A notification should pop up asking you about the environment. If not, click on the v1.9 environment symbol in the bottom bar of VSCode, or, alternatively, open the command palette (Ctrl/Cmd + Shift + P), start typing julia, and select Julia: Change current environment. Set the environment to NeuralClosure.

6. In the file explorer (Ctrl/Cmd + Shift + E), you should see the files. Open tutorials/burgers.ipynb. You can skip the Colab section. Start with uncommenting and executing the cell with Pkg.instantiate(), which will install the local dependencies. Execute the notebook one cell at the time with Shift + Enter. Output and plots should appear below each cell.

Note: If you do not like the notebook format, you can alternatively open the source script tutorials/burgers.jl or the markdown file tutorials/burgers.md. Both can be executed interactively one Julia line at the time with Shift + Enter. Then a separate Julia REPL will pop up, and plots will appear in a separate plot pane. To visualize the LaTeX equations, open the markdown file and press Ctrl/Cmd + Shift + V to render the markdown (with equations) in a new pane.

Option 2: Running on Google Colab

If you do not want to install Julia locally, you can run it on a Google cloud machine instead. This requires a Google account. Steps:

1. Open Google Colab
2. Sign in to your Google account.
3. Click on File > Upload Notebook > GitHub and paste the following url:

   https://github.com/agdestein/NeuralClosure/
   

   Select tutorials/burgers.ipynb.
4. Follow the Colab instructions in the top of the notebook.