📊 Parameter Estimation for differential equations with Julia

This lecture is part of the lecture series organized by the Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University.

📑 Summary

Statistics and optimisation theory provide rich theories to find optimal model parameters for given datapoints. These methodologies adapt well to ODE and PDE models. Depending on the number of unknown parameters and available data, different strategies are optimal. We will discuss optimisation based parameter estimation and automatic differentiation, a technique to obtain accurate gradients.

Figure: Example parameter fitting using an $L^2$ error and a quasi-Newton (LBFGS) solver.

📚 Material

Lecture notes and example code (links in preparation).

💻 Setup instructions

This lecture will be interactive with various coding sessions. Please install Julia in advace! 👍

Please install:

The Julia programming language,
Visual Studio Code,
Julia for Visual Studio Code,
Clone github.com/SteffenPL/ParamEstimationLecture2024,
Install all required Julia dependencies^[1] either by running the file 000_setup.jl or with the following steps:
- Open the repository in VS Code and Open Julia REPL^[2],
- enter package mode by pressing ] inside the Julia REPL and
- run activate . and instantiate.

1. Installation of the Julia programming language

We will follow the official installation instructions from julialang.org/downloads.

For Windows, open a terminal and type
```
winget install julia -s msstore
```

For Linux/MacOS, open a terminal and type

curl -fsSL https://install.julialang.org | sh

To test the installation, you can type julia into the terminal, which should start the most recent version of Julia!

2. Installation of VS Code and the Julia extensions

Visual Studio Code is a modern text editor^[3] which can function as an IDE for many programming languages. It is the current de-facto standard for programming in Julia.

We will follow the instructions from code.visualstudio.com/docs/languages/julia.

Install VS Code for your platform: https://code.visualstudio.com/download.
Open the Julia extension on the VS Code Marketplace and press Install; or manually install by doing the following steps:
- Start VS Code.
- Inside VS Code, go to the Extensions view by clicking View on the top menu bar and then selecting Extensions.
- In the Extensions view, search for the term "julia" in the Marketplace search box, then select the Julia extension (julialang.language-julia) and select the Install button.
- Restart VS Code.

For basic operations inside VS Code, please check code.visualstudio.com/docs/getstarted/getting-started.

The most useful command in Visual Studio Code is the command palette, where you can search for any commands!

You can access it with the shortcut: Shift + Command + P (Mac) or Ctrl + Shift + P Windows/Linux.
- For example: Search for Preferences: Color Theme to select a nice theme!

3. Installing Julia dependencies

Download the lectures example code from github.com/SteffenPL/ParamEstimationLecture2024.
Open the folder with Visual Studio Code File > Open Folder.^[4]
Open the file 000_setup.jl and press the run button in the top right corner of the editor (or run the command Julia: Execute Active File in REPL^[2:1]). This should download and install all requirements.
To test the installation, open the file 002_first_example.jl and also run this file. It should display a plot of an ODE solution.^[5]

4. Running Julia Code

You can also run Julia code by selecting a few lines in the editor and using the shortcut Shift+Enter.

5. A note on installing Julia packages

One of the best features of Julia is the large ecosystem of very modern scientific computing packages. One can see a great overview at the documentation side of the SciML project (Scientific Machine Learning).

To add a new package to your current code, you can use the integrated Julia package manager as follows:

Open a Julia REPL.
Press the ] key to enter the package mode.
Optional but recommended: Activate a new Julia enviroment for your current folder with the command activate . (where . denotes the current folder).
Use the command add to install a new package, for example via

add OrdinaryDiffEq Plots

to install the two packages OrdinaryDiffEq and Plots.

Once you added a package, you and use it's functions with the Julia command

using OrdinaryDiffEq, Plots

inside your Julia code.

💬 Organisation

This lecture is given in English.
It is part of the lecture series organized by the Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University.
This lecture series is supported by
- JSPS KAKENHI Grant Number JP23H04936, 23K13013, 24H00188
- JST CREST Grant Number JPMJCR1926
- JST Moonshot Program Grant Number JPMJMS2023

🔗 Useful links

Julia Tutorial: docs.julialang.org/en/v1
Cheat-Sheet: steffenpl.github.io/Julia-Cheat-Sheet
Modern Julia: https://modernjuliaworkflows.org/
Noteworthy differences (from MATLAB/Python/C...): docs.julialang.org/en/v1/manual/noteworthy-differences
Very friendly discussion forum with some strong mathematicians as regulars: discourse.julialang.org

The dependencies might change shortly before the lecture. If any problems arise, please contact me. ↩︎
To find any command in Visual Studio code, open the command palette via the shortcut Shift + Command + P (Mac) or Ctrl + Shift + P Windows/Linux. ↩︎ ↩︎
If you prefer strict free open-source software, one might prefer VSCodium. ↩︎
The first time you open a new folder, Visual Studio Code might ask you to confirm that you trust the files. This is normal behaviour and in order to work on a program you need to agree to trust the folder: ↩︎
The first runtime might be slow, due to one-time precompilation. ↩︎