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