# **Function Discovery** for **Oridinary Differential Equation**

#### **Problem** 
The Ordinary Differential Equation (ODE) Function Discovery problem focuses on identifying functional relationships by minimizing mean square error using given X values and constant parameters.

```{image} ./ode.png
:width: 80%
:align: center
```

+ **Given:** X value, a set of constant parameters.

+ **Objective:** Minimize the mean square error.

+ **Constraints:** 
    - None


#### Algorithm Design Task

+ **The task** is to design the function to fit the dataset.
  + **Inputs:** X value.
  + **Outputs:** Predicted y value.

#### Evaluation

+ **Dataset:** Dataset from **ODEFormer: Symbolic Regression of Dynamical Systems with Transformers**. 

+ **Fitness:** Mean Square Error


#### Template: 

```python
template_program = '''
import numpy as np

def equation(x: float, params: np.ndarray) -> float:
    """ A ODE mathematical function    
    Args:
        x: the initial float value of the ode formula
        params: a 1-d Array of numeric constants or parameters to be optimized

    Return:
        A numpy array representing the result of applying the mathematical function to the inputs.
    """
    y = params[0] * x + params[2]
    return y


'''

task_description = "Find the ODE mathematical function skeleton, given data on initial x. The function should be differentiable, continuous. Only selectable components: 1. Basic operators: +, -, *, /, ^, np.sqrt, np.exp, np.log, np.abs 2. Trigonometric expressions: np.sin, np.cos, np.tan, np.arcsin, np.arccos, np.arctan 3. Standard constants: 'np.pi' represents pi and 'np.e' represents Euler's number"



```