enhance: tutorial 1 using ann.

[kshitij-maths]

Overview

This PR adds a new tutorial (tutorial-1-ann.ipynb) that demonstrates the construction of a Reduced Order Model (ROM) using Artificial Neural Networks (ANN) instead of the traditional Radial Basis Function (RBF) approach.

The tutorial follows the workflow established in the original Tutorial 1 but the RBF is replaced with an ANN.

Key Changes

• New Tutorial: Created tutorial-1-ann.ipynb showcasing the ANN-ROM workflow.
• Performance Comparison: Included a new comparative analysis section between ANN and RBF. This demonstrates the ANN's better accuracy (measured via relative $L^2$ error) on this specific dataset.

Technical Details

• POD Rank: Maintained a consistent rank (n_snapshots - 1) across both the ANN and RBF models to ensure a comparable and fair benchmark.
• Metrics: Utilized Leave-One-Out (LOO) Cross-Validation to assess the L2 relative error.
• Visualization: Added log-scale error plots to visualize the performance gap between the two interpolation methods.

Verification Results

On the provided thermal dataset, the ANN achieved significantly lower error rates compared to the RBF:

• Mean LOO Error (ANN): ~10⁻² to 10⁻³
• Mean LOO Error (RBF): ~10⁻¹

Related Issue

This PR fixes issue #179

[ndem0]

Dear @kshitij-maths, thanks!
The tutorial is fine, but there are several things I do not understand:

• tutorial-2 already shows the usage of ANN (and also AE). I'm a little bit lost in understanding the take-home message: what are the additional information can I get from this tutorial?
• Why is the name tutorial-1-ann? Shouldn't it be tutorial-5?
• The link is missing in the README
• check the tutorial is also present in the documentation files
• please use the data from mathlab huggingface https://huggingface.co/SISSAmathLab

[kshitij-maths]

Dear @ndem0, thanks!

• Yes, you're right. I also observed that but I did this as the comment ( ENH request: create also a tutorial with ANN #179 (comment)) specifically asked this.
• We can do this, but I found it more intuitive. Because it is not a completely new tutorial but just tutorial-1 with a different interpolator.
• Do you mean tutorials/README? Thanks, will update.
• Will check.
• Okay.

[ndem0]

Dear @kshitij-maths, why did you change the .rst files?

[kshitij-maths]

Dear @ndem0, I did that to include it in the documentation.

• check the tutorial is also present in the documentation files

[ndem0]

No, you actually changed the structure of (some of) the documentation files, in particular:

database.rst
parameter.rst
reducedordermodel.rst
regular_grid.rst
snapshot.rst

Those files don't control the presence of the tutorials in the documentation, they actually change the documentation of these classes. Please revert it. The only file to change is tutorials.rst.

[kshitij-maths]

Yes, you're right. Is it okay now?