Participants in this competition will be evaluated based on the accuracy of their Physics-Informed Neural Network (PINN) models in solving the given Convection-Diffusion (CD) equation. The evaluation process is split into two phases with separate leaderboards:

---

# 1. Public Leaderboard Evaluation

## During the competition:
- Participants will submit predictions for a provided set of **test points( \\(N_{test}\\) test points will be provided)** specified in `test.csv`.  
- These points represent coordinates \\((x, y)\\) within the domain \\([0,1] \times [0,1]\\).  
- The predicted values will be compared to a **reference solution** of the PDE at these points.  

### Submission Format
The submission file should contain the following columns:
- `ID`: A unique identifier for each row. It is important that this column contains unique integers starting from 1. The ID column is used to correlate rows in the test set with the submission data.
- `x`: The x-coordinate of the test point.
- `y`: The y-coordinate of the test point.
- `u`: Your predicted solution value at the \\((x, y)\\) coordinate, generated by your model.

The order of the points in the `test.csv` must be preserved in your submission. Ensure that your `u` values align with the correct `(x, y)` coordinates in the same order as provided in `test.csv`.

### Example of the Submission Format:

| ID  | x    | y    | u     |
|-----|------|------|-------|
| 1   | 0.1  | 0.2  | 0.123 |
| 2   | 0.3  | 0.4  | 0.234 |
| 3   | 0.5  | 0.6  | 0.345 |
| ... | ...  | ...  | ...   |

### Important Notes:
- The **ID** column must be included in the submission file as it is required for scoring purposes.
- The **order of the rows** in your submission must match the order of the test points provided in `test.csv`. If the order is different, the evaluation script will not be able to match the correct test points to your predictions.

Ensure that your submission adheres to this format, or it will not be accepted. Any deviation from this format may result in disqualification or errors during evaluation.

### **Metric**: \\(l_2\\) Error (Root Mean Square Error)  
The \\(l_2\\) error will be calculated as follows:

$$
l_2 = \sqrt{\frac{1}{N_{test}} \sum_{i=1}^{N_{test}} \left( u_{\text{pred}}(x_i, y_i) - u_{\text{true}}(x_i, y_i) \right)^2}
$$

Where:  
- \\(N_{test}\\): Number of test points in `test.csv`  
- \\(u_{\text{pred}}(x_i, y_i)\\): Predicted solution at test point \\((x_i, y_i)\\)  
- \\(u_{\text{true}}(x_i, y_i)\\): Exact solution of the PDE at test point \\((x_i, y_i)\\)  

---

# **2. Private Leaderboard Evaluation**

At the conclusion of the competition:
- The **top 10 participants** from the public leaderboard will have their models evaluated on a **sample drawn from the given test points**.  
- These points will be selected from the same domain \\([0,1]\times[0,1]\\).  
- The \\(l_2\\) error will again be used as the evaluation metric on this private test set.  

The **private leaderboard** will determine the final rankings and winners of the competition.

---

# **Important Notes**  

- **Focus on Generalization**: Ensure that your PINN model generalizes well to unseen data points. Overfitting to the public test set may result in poor performance on the private test set.  
- **Tiebreaker**: In case of a tie in \\(l_2\\) error, submissions will be ranked based on the submission timestamp, with earlier submissions given higher priority.  

Are you ready to take on the challenge and build a robust, generalizable solution? Let's predict and prevent environmental disasters with the power of scientific machine learning! 🚀