scimba_torch.physical_models.temporal_pde.heat_equation

Heat equation in strong form.

Functions

zeros_bc_rhs(w, t, x, n, mu[, nb_func])	Function returning a zero right-hand side for the boundary conditions.
zeros_rhs(w, t, x, mu[, nb_func])	Function returning a zero right-hand side.

Classes

HeatEquation1DDirichletStrongForm(space, init)	Implementation of a 1D heat equation with Dirichlet BCs in strong form.
HeatEquation1DStrongForm(space, init[, f, g])	Implementation of a 1D heat equation with Neumann BCs in strong form.
HeatEquation2DStrongForm(space, init[, f, g])	Implementation of a 2D Laplacian problem with Dirichlet BCs in strong form.
HeatEquation2DStrongFormImplicit(space, init)	2D heat equation for implicit discrete_pinns.

zeros_rhs(w, t, x, mu, nb_func=1)

Function returning a zero right-hand side.

Parameters:

• w (MultiLabelTensor) – Solution tensor.

• t (LabelTensor) – Temporal coordinates tensor.

• x (LabelTensor) – Spatial coordinates tensor.

• mu (LabelTensor) – Parameter tensor.

• nb_func (int) – Number of functions to return (default is 1).

Return type:

Tensor

Returns:

A tensor of zeros with shape (number of points, nb_func).

zeros_bc_rhs(w, t, x, n, mu, nb_func=1)

Function returning a zero right-hand side for the boundary conditions.

Parameters:

• w (MultiLabelTensor) – Solution tensor.

• t (LabelTensor) – Temporal coordinates tensor.

• x (LabelTensor) – Spatial coordinates tensor.

• n (LabelTensor) – Normal vector tensor.

• mu (LabelTensor) – Parameter tensor.

• nb_func (int) – Number of functions to return (default is 1).

Return type:

Tensor

Returns:

A tensor of zeros with shape (number of points, nb_func).

class HeatEquation1DStrongForm(space, init, f=<function zeros_rhs>, g=<function zeros_bc_rhs>, **kwargs)

Bases: FirstOrderTemporalPDE

Implementation of a 1D heat equation with Neumann BCs in strong form.

Parameters:

• space (AbstractApproxSpace) – The approximation space for the problem

• init (Callable) – Callable for the initial condition

• f (Callable) – Source term function (default is zero)

• g (Callable) – Neumann boundary condition function (default is zero)

• **kwargs – Additional keyword arguments

space_operator(w, t, x, mu)

Apply the spatial operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor | tuple[Tensor, ...]

Returns:

Spatial operator tensor

bc_operator(w, t, x, n, mu)

Apply the boundary condition operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition operator tensor

rhs(w, t, x, mu)

Compute the right-hand side (RHS) of the PDE.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

RHS tensor

bc_rhs(w, t, x, n, mu)

Compute the boundary condition RHS.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition RHS tensor

initial_condition(x, mu)

Compute the initial condition.

Parameters:

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Initial condition tensor

functional_operator(func, t, x, mu, theta)

Compute the functional operator.

Parameters:

• func (VarArgCallable) – Callable representing the function

• t (Tensor) – Temporal coordinate tensor

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor

functional_operator_bc(func, t, x, n, mu, theta)

Compute the functional operator for boundary conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• t (Tensor) – Temporal coordinate tensor

• x (Tensor) – Spatial coordinate tensor

• n (Tensor) – Normal vector tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for boundary conditions

functional_operator_ic(func, x, mu, theta)

Compute the functional operator for initial conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for initial conditions

class HeatEquation1DDirichletStrongForm(space, init, f=<function zeros_rhs>, g=<function zeros_bc_rhs>, **kwargs)

Bases: FirstOrderTemporalPDE

Implementation of a 1D heat equation with Dirichlet BCs in strong form.

Parameters:

• space (AbstractApproxSpace) – The approximation space for the problem

• init (Callable) – Callable for the initial condition

• f (Callable) – Source term function (default is zero)

• g (Callable) – Dirichlet boundary condition function (default is zero)

• **kwargs – Additional keyword arguments

space_operator(w, t, x, mu)

Apply the spatial operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Spatial operator tensor

bc_operator(w, t, x, n, mu)

Apply the boundary condition operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition operator tensor

rhs(w, t, x, mu)

Compute the right-hand side (RHS) of the PDE.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

RHS tensor

bc_rhs(w, t, x, n, mu)

Compute the boundary condition RHS.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition RHS tensor

initial_condition(x, mu)

Compute the initial condition.

Parameters:

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Initial condition tensor

functional_operator(func, t, x, mu, theta)

Compute the functional operator.

Parameters:

• func (VarArgCallable) – Callable representing the function

• t (Tensor) – Temporal coordinate tensor

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor

functional_operator_bc(func, t, x, n, mu, theta)

Compute the functional operator for boundary conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• t (Tensor) – Temporal coordinate tensor

• x (Tensor) – Spatial coordinate tensor

• n (Tensor) – Normal vector tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for boundary conditions

functional_operator_ic(func, x, mu, theta)

Compute the functional operator for initial conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for initial conditions

class HeatEquation2DStrongForm(space, init, f=<function zeros_rhs>, g=<function zeros_bc_rhs>, **kwargs)

Bases: FirstOrderTemporalPDE

Implementation of a 2D Laplacian problem with Dirichlet BCs in strong form.

Parameters:

• space (AbstractApproxSpace) – The approximation space for the problem

• init (Callable) – Callable for the initial condition

• f (Callable) – Source term function (default is zero)

• g (Callable) – Dirichlet boundary condition function (default is zero)

• **kwargs – Additional keyword arguments

space_operator(w, t, x, mu)

Apply the spatial operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Spatial operator tensor

bc_operator(w, t, x, n, mu)

Apply the boundary condition operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition operator tensor

rhs(w, t, x, mu)

Compute the right-hand side (RHS) of the PDE.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

RHS tensor

bc_rhs(w, t, x, n, mu)

Compute the boundary condition RHS.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition RHS tensor

initial_condition(x, mu)

Compute the initial condition.

Parameters:

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Initial condition tensor

class HeatEquation2DStrongFormImplicit(space, init, f=<function zeros_rhs>, g=<function zeros_bc_rhs>, **kwargs)

Bases: FirstOrderTemporalPDE

2D heat equation for implicit discrete_pinns.

Parameters:

• space (AbstractApproxSpace) – the approx. space.

• init (Callable) – the rhs of the initial condition.

• f (Callable) – the rhs of the residual (default is zero).

• g (Callable) – the rhs of the boundary condition (default is zero).

• **kwargs – Additional keyword arguments.

space_operator(w, t, x, mu)

Apply the spatial operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Spatial operator tensor

bc_operator(w, t, x, n, mu)

Apply the boundary condition operator.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition operator tensor

rhs(w, t, x, mu)

Compute the right-hand side (RHS) of the PDE.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

RHS tensor

bc_rhs(w, t, x, n, mu)

Compute the boundary condition RHS.

Parameters:

• w (MultiLabelTensor) – Solution tensor

• t (LabelTensor) – Temporal coordinate tensor

• x (LabelTensor) – Spatial coordinate tensor

• n (LabelTensor) – Normal vector tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Boundary condition RHS tensor

initial_condition(x, mu)

Compute the initial condition.

Parameters:

• x (LabelTensor) – Spatial coordinate tensor

• mu (LabelTensor) – Parameter tensor

Return type:

Tensor

Returns:

Initial condition tensor

functional_operator(func, x, mu, theta)

Compute the functional operator.

Parameters:

• func (VarArgCallable) – Callable representing the function

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor

functional_operator_bc(func, t, x, n, mu, theta)

Compute the functional operator for boundary conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• t (LabelTensor) – Temporal coordinate tensor

• x (Tensor) – Spatial coordinate tensor

• n (Tensor) – Normal vector tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for boundary conditions

functional_operator_ic(func, x, mu, theta)

Compute the functional operator for initial conditions.

Parameters:

• func (VarArgCallable) – Callable representing the function

• x (Tensor) – Spatial coordinate tensor

• mu (Tensor) – Parameter tensor

• theta (Tensor) – Additional parameters tensor

Return type:

Tensor

Returns:

Functional operator tensor for initial conditions