Update Tutorials (#544)

* update tutorials
* tutorial guidelines
* doc

tutorials/TUTORIAL_GUIDELINES.md

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+# PINA Tutorial Guidelines
+
+Welcome to the **PINA Tutorial Guidelines** — a guiding document that defines the structure, style, and pedagogical philosophy for all tutorials in the **PINA** package. The goal of this guideline is to ensure that all learning materials are **clear, consistent, pedagogically sound, and beginner-friendly**, while remaining powerful enough to support advanced use cases.
+
+
+## Purpose
+
+The purpose of the PINA tutorials is to help users:
+
+- Gaining a solid understanding of the PINA library and its core functionalities.
+- Learning how to work with the PINA modules.
+- Explore practical and advanced applications using consistent, hands-on code examples.
+
+
+## Guiding Principles
+
+1. **Clarity Over Cleverness**  
+   Tutorials should aim to teach, not impress. Prioritize readable and understandable code and explanations.
+
+2. **Progressive Disclosure of Complexity**  
+   Start simple and gradually introduce complexity. Avoid overwhelming users early on.
+
+3. **Consistency is Key**  
+   All tutorials should follow a common structure (see below), use the same markdown and code formatting, and have a predictable flow.
+
+4. **Real Applications, Real Problems**  
+   Ground tutorials in real Scientific Applications or datasets, wherever possible. Bridge theory and implementation.
+
+
+## Tutorial Structure
+
+To ensure clarity, consistency, and accessibility, all PINA tutorials should follow the same standardized format.
+
+### 1. Title
+
+Each tutorial must begin with a clear and descriptive title in the following format: **Tutorial: TUTORIAL_TITLE**. The title should succinctly communicate the focus and objective of the tutorial.
+
+### 2. Introducing the Topic
+
+Immediately after the title, include a short introduction that outlines the tutorial's purpose and scope.
+
+- Briefly explain what the tutorial covers and why it’s useful.
+- Link to relevant research papers, publications, or external resources if applicable.
+- List the core PINA components or modules that will be utilized.
+
+### 3. Imports and Setup
+
+Include a Python code cell with the necessary setup. This ensures that the tutorial runs both locally and on platforms like Google Colab.
+
+```python
+## Routine needed to run the notebook on Google Colab
+try:
+    import google.colab
+    IN_COLAB = True
+except:
+    IN_COLAB = False
+
+if IN_COLAB:
+    !pip install "pina-mathlab[tutorial]"
+
+import torch                        # if used
+import matplotlib.pyplot as plt     # if used
+import warnings                     # if needed
+
+warnings.filterwarnings("ignore")
+
+# Additional PINA and problem-specific imports
+...
+```
+
+### 3. Data Generation or Loading
+* Describe how the data is generated or loaded.
+* Include commentary on data structure, format, and content.
+* If applicable, visualize key features of the dataset or simulation domain.
+
+### 4. Main Body
+The core section of the tutorial should present the problem-solving process in a clear, structured, and pedagogical way. This is where the tutorial delivers the key learning objectives.
+
+- Guide the user step-by-step through the PINA workflow.
+- Introduce relevant PINA components as they are used.
+- Provide context and explain the rationale behind modeling decisions.
+- Break down complex sections with inline comments and markdown explanations.
+- Emphasize the relevance of each step to the broader goal of the tutorial.
+
+### 5. Results, Visualization and Error Analysis
+- Show relevant plots of results (e.g., predicted vs. ground truth).
+- Quantify performance using metrics like loss or relative error.
+- Discuss the outcomes: strengths, limitations, and any unexpected behavior
+
+### 6. What's Next?
+All the tutorials are concluded with the **What's Next?** section,giving suggestions for further exploration. For this use the following format:
+```markdown
+## What's Next?
+
+Congratulations on completing the ..., here are a few directions you can explore:
+
+1. **Direction 1** — Suggestion ....
+
+2. **Direction 2** — Suggestion ....
+
+3. **...and many more!** — Other suggestions ....
+
+For more resources and tutorials, check out the [PINA Documentation](https://mathlab.github.io/PINA/).
+```
+
+## Writing Style
+
+- Use **clear markdown headers** to segment sections.
+- Include **inline math** with `$...$` and display math with `$$...$$`.
+- Keep paragraphs short and focused.
+- Use **bold** and *italic* for emphasis and structure.
+- Include comments in code for clarity.
+
+
+## Testing Tutorials
+
+Every tutorial should:
+- Be executable from top to bottom.
+- Use the `tutorial` requirements in the [`pyproject.toml`](https://github.com/mathLab/PINA/blob/6ed3ca04fee3ae3673d53ea384437ce270f008da/pyproject.toml#L40) file.
+
+
+## Contributing Checklist
+
+We welcome contributions! If you’re writing a tutorial:
+1. The tutorial follows this guidelines for structure and tone.
+2. The tutorial is simple and modular — one tutorial per concept.
+3. The tutorial PRs contains only the `.ipynb` file, and the updated `README.md` file.
+