# Finetune the LLM: Interpret Heat Equation Solution — Data

来源：https://www.kaggle.com/competitions/fine-tuning-lm-physical-interpretation-hackathon/data

本地 Meta Kaggle 快照；不是本次实时页面。

# Dataset Description

For validation, we are providing solution of heat equation for very simple cases. Participants have to create their own training dataset for finetuning (for temporal cases and different conductivity values with different boundary conditions as well). 
The time dependent heat equation is given as 
$$\dfrac{\partial T}{\partial t} - \alpha\Big(\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\Big)T = f(x,y,t)$$
where 
- \\(T\\) : Temperature field  
- \\(\alpha\\) : Thermal conductivity  
- \\(f\\) : Force function
## Details of the solution for different cases on [0,1] X [0,1] sqaure : 
- **Case-1**  
```
∂²T/∂x² + ∂²T/∂y² = 8π²sin(2πx)sin(2πy)
```
*Boundary Conditions*
```
T(0,y) = 0
T(1,y) = 0
T(x,0) = 0
T(x,1) = 0
```

- **Case-2**  
```
∂²T/∂x² + ∂²T/∂y² = 0  
```
*Initial and Boundary Conditions*
```
T(0,y) = 0
T(1,y) = 0
T(x,0) = 0
T(x,1) = sin(πx)
```

- **Case-3**
```
∂²T/∂x² + ∂²T/∂y² = 0
```
*Boundary Conditions*
```
T(0,y) = 0
T(1,y) = y(1-y)
T(x,0) = 0
T(x,1) = 0
```

- **Case-4**
```
∂T/∂t = α(∂²T/∂x² + ∂²T/∂y²)
```
*Initial and Boundary Conditions*
```
T(x,y,0) = sin(πx)sin(πy)
T(0,y,t) = 0
T(1,y,t) = 0
T(x,0,t) = 0
T(x,1,t) = 0
α = 0.01
```
## Example of the Submission Format:

| ID  | Answer    |
|-----|------|
| 1   | - | 
| 2   | - |
| 3   | - |
| ... | ... | 
| 12 | - |
- In the answer column, the participants need to put the fine-tuned model's output corresponding to the question ID in Questions.csv.   
- Answer to the questions should be in the correct order.
- Name of the submission file should be **submission.csv**

## Files

*   **Case1.vtk** - vtk file with the coordinates of domain and solution of PDE at those coordinates for Case-1 on unit square mesh
*   **Case2.vtk, Case3.vtk, Case4.vtk**
*   **Questions.csv** - csv file with the questions for each Case to be asked while giving prompt to fine-tuned LLM
*   **Sample_submission.csv** - a sample submission file in the correct format.