scimba_torch.numerical_solvers.temporal_pde.neural_semilagrangian

Implement the Neural Semi-Lagrangian scheme.

Classes

Characteristic(pde, **kwargs)	Class to handle the characteristics of a PDE.
NeuralSemiLagrangian(characteristic, ...)	Implement the Neural Semi-Lagrangian scheme.

class Characteristic(pde, **kwargs)

Bases: object

Class to handle the characteristics of a PDE.

Parameters:

• pde (AdvectionReactionDiffusion) – The PDE for which to compute characteristics.

• **kwargs – Additional keyword arguments.

make_diffusion_directions(kind='directionwise')

Creates the diffusion directions based on the chosen kind.

Parameters:

kind (str) – The kind of diffusion directions to create (Default: “directionwise”).

Return type:

Tensor

Returns:

The created diffusion directions.

ensure_periodicity(y)

Ensures that the points x are within the domain bounds.

If the points are outside the bounds, they are wrapped around the domain.

Parameters:

y (Tensor) – The points.

Returns:

The points within the domain bounds.

exact_foot_constant_advection(t, x, mu, dt)

Computes the foot of the (backwards) characteristic curve.

The curve is originating from point x at time t + dt, towards point y at time t.

In the case of constant advection, the foot is given by y = x - a * dt.

Parameters:

• t (Tensor) – The current time.

• x (Tensor) – The current points.

• mu (Tensor) – The physical parameters.

• dt (float) – The time step.

Return type:

Tensor

Returns:

The foot of the characteristic curve.

static backwards_ode_integrator(f, t, x, mu, dt, scheme='rk4')

Computes the solution of the ODE dx/dt = f(t, x, mu) at time t - dt.

Use a numerical scheme.

Parameters:

• f (Callable) – The function defining the ODE.

• t (Tensor) – The current time.

• x (Tensor) – The current points.

• mu (Tensor) – The physical parameters.

• dt (float) – The time step.

• scheme (str) – The numerical scheme used for integrating the ODE. Default is “rk4”.

Return type:

Tensor

Returns:

The numberical solution at time t - dt.

numerical_foot(t, x, mu, dt, **kwargs)

Computes the foot of the (backwards) characteristic curve.

The curve is originating from point x at time t + dt, towards point y at time t.

In the case of non-constant advection, the foot is computed using a numerical scheme, with n_sub_steps sub-steps.

Parameters:

• t (Tensor) – The current time.

• x (Tensor) – The current points.

• mu (Tensor) – The physical parameters.

• dt (float) – The time step.

• **kwargs –

  Additional keyword arguments including:

  • scheme (str): The numerical scheme used for integrating the ODE. Default is “rk4”.

  • n_sub_steps (int): The number of sub-steps used for integrating the ODE. Default is 5.

Returns:

The foot of the characteristic curve.

Return type:

torch.Tensor

get_foot(t, x, mu, dt, **kwargs)

Computes the foot of the (backwards) characteristic curve.

The curve is originating from point x at time t + dt, towards point y at time t.

If an exact foot is provided, it is used. Otherwise, a numerical foot is computed.

Periodic and flipped periodic boundary conditions are supported.

Parameters:

• t (LabelTensor) – The current time.

• x (LabelTensor) – The current points.

• mu (LabelTensor) – The physical parameters.

• dt (float) – The time step.

• **kwargs –

  Additional keyword arguments including:

  • scheme (str): The numerical scheme used for integrating the ODE, in case no exact foot is provided. Default is “rk4”.

  • n_sub_steps (int): The number of sub-steps used for integrating the ODE, in case no exact foot is provided. Default is 5.

Return type:

LabelTensor

Returns:

The foot of the characteristic curve.

get_foot_kinetic_pde(t, x, v, mu, dt, **kwargs)

Computes the foot of the (backwards) characteristic curve for a kinetic PDE.

Use get_foot.

Parameters:

• t (LabelTensor) – The current time.

• x (LabelTensor) – The current points.

• v (LabelTensor) – The velocity points.

• mu (LabelTensor) – The physical parameters.

• dt (float) – The time step.

• **kwargs –

  Additional keyword arguments including:

  • scheme (str): The numerical scheme used for integrating the ODE, in case no exact foot is provided. Default is “rk4”.

  • n_sub_steps (int): The number of sub-steps used for integrating the ODE, in case no exact foot is provided. Default is 5.

Return type:

LabelTensor

Returns:

The foot of the characteristic curve.

class NeuralSemiLagrangian(characteristic, projector, **kwargs)

Bases: ExplicitTimeDiscreteScheme

Implement the Neural Semi-Lagrangian scheme.

Parameters:

• characteristic (Characteristic) – The characteristics model.

• projector (AbstractNonlinearProjector) – The projector for training the model.

• **kwargs – Additional hyperparameters for the scheme.

construct_rhs(pde_n, t, dt, **kwargs)

Constructs the RHS of the Neural Semi-Lagrangian scheme.

Computes the foot of the characteristic curve originating from point x at time t + dt, towards point y at time t.

Parameters:

• pde_n (AdvectionReactionDiffusion) – The PDE to solve at the current time step.

• t (float) – The current time.

• dt (float) – The time step.

• **kwargs –

  Additional keyword arguments including:

  • scheme (str): The numerical scheme used for integrating the ODE, in case no exact foot is provided. Default is “rk4”.

  • n_sub_steps (int): The number of sub-steps used for integrating the ODE, in case no exact foot is provided. Default is 5.

Return type:

Callable

Returns:

The RHS function for the Neural Semi-Lagrangian scheme.

update(t, dt, **kwargs)

Computes the next time step of the Neural Semi-Lagrangian method.

Parameters:

• t (float) – Current time.

• dt (float) – Time step.

• **kwargs – Additional keyword arguments.