Heat Conduction Model

class Heat(central_param: ndarray = array([1.5, 1.5, 0.5]), param_limits: ndarray = array([[1., 2.], [1., 2.], [0., 1.]]), name: str | None = None, **kwargs)

Bases: JaxModel

A two-dimensional anisotropic heat conduction equation model on a square domain with four dirichlet boundaries.

The model is defined by the following partial differential equation on the square spacial domain \(\Omega = [0, 1]^2\) on the time interval \([0, 0.1]\):

\[\frac{\partial u}{\partial t} = \nabla \cdot \left( \kappa \nabla u \right)\]

subject to

\[u(x, y, t=0) = 0\]

and

\[u(0, y, t) = 1, \quad u(1, y, t) = 0, \quad u(x, 0, t) = 1, \quad u(x, 1, t) = 0\]

with the symmetric positive definite thermal conductivity matrix \(\kappa\) and the temperature \(u\). Inference is performed on the entries of \(\kappa\): param[0] = \(\kappa_{11}\), param[1] = \(\kappa_{22}\), param[2] = \(\kappa_{12}\).

classmethod forward(param: ndarray) → ndarray

Forward method for the heat model.

Yields the solution of the anisotropic heat conduction equation at time \(t=0.1\) in five spatial points, which are arranged similar to the number “five” on a dice.

Parameters:

param (np.ndarray) – Entries of the conductivity matrix: param[0] = \(\kappa_{11}\), param[1] = \(\kappa_{22}\), param[2] = \(\kappa_{12}\)

Returns:

The solution of the anisotropic heat conduction equation at time \(t=0.1\)

Return type:

np.ndarray

in five spacial points, which are arranged similar to the number “five” on a dice.

param_is_within_domain(param: ndarray) → bool

Checks whether a parameter is within the parameter domain of the model. This condition stems from thermodynamical considerations.

Parameters:

param (np.ndarray) – The parameter to check.

Returns:

True if the parameter is within the limits.

Return type:

bool

classmethod perform_simulation(kappa: ndarray) → ndarray

Performs a simulation of the heat equation with the given parameters.

Parameters:

kappa (np.ndarray) – Entries of the conductivity matrix: kappa[0] = \(\kappa_{11}\), kappa[1] = \(\kappa_{22}\), kapp[2] = \(\kappa_{12}\)

Returns:

An array containing the solution of the anisotropic heat conduction equation, where the first two indices correspond to the x and y coordinates, respectively. The time is fixed to the class variable t_end.

Return type:

np.ndarray

CENTRAL_PARAM = array([1.5, 1.5, 0.5])

PARAM_LIMITS = array([[1., 2.],        [1., 2.],        [0., 1.]])

data_dim: int | None = 5

The values of the heat equation solution at five points are observed. See evaluation_points.

evaluation_points = Array([[0.25, 0.25],        [0.75, 0.25],        [0.5 , 0.5 ],        [0.25, 0.75],        [0.75, 0.75]], dtype=float64)

num_grid_points = 20

Number of grid points in each spatial direction

param_dim: int | None = 3

Number of independent heat conductivity parameters

plate_length = Array([1., 1.], dtype=float64)

Length of the square plate

t_end = 0.1

End time of the simulation

class HeatArtificial(central_param: ndarray = array([1.5, 1.5, 0.5]), param_limits: ndarray = array([[1., 2.], [1., 2.], [0., 1.]]), name: str | None = None, **kwargs)

Bases: Heat, ArtificialModelInterface

generate_artificial_params(num_samples: int, independent_params: bool = True) → ndarray

Generates a set of viable parameter samples for the heat model.

Parameters:

• num_samples (int) – Number of samples to generate.

• independent_params (bool, optional) – Whether the parameters should be independent. Defaults to True.

Returns:

A set of viable parameter samples.

Return type:

np.ndarray

CENTRAL_PARAM = array([1.5, 1.5, 0.5])

PARAM_LIMITS = array([[1., 2.],        [1., 2.],        [0., 1.]])

num_grid_points = 20

Number of grid points in each spatial direction

heat_rhs(t: float, u: Array, args: tuple | list) → Array

Right hand side of the heat equation.

Parameters:

• t (float) – time at which the right hand side is evaluated

• u (jnp.ndarray) – current solution of the heat equation

• args (tuple | list) – tuple of the form (dx, dy, param) with dx, dy the spatial discretization and param the thermal conductivity matrix, given as a vector of length 3 with param[0] = kappa_11, param[1] = kappa_22, param[2] = kappa_12

Returns:

right hand side of the heat equation

Return type:

jnp.ndarray