scimba_torch.numerical_solvers.temporal_pde.time_discrete¶

Time-discrete numerical solvers for temporal PDEs.

Classes

`ExplicitTimeDiscreteScheme`(pde, projector[, ...])	An explicit time-discrete scheme for solving a differential equation.
`TimeDiscreteAnagramProjector`(pde, **kwargs)	A time-discrete natural gradient projector for solving temporal PDEs.
`TimeDiscreteCollocationProjector`(pde, **kwargs)	Implement the Galerkin-based nonlinear projection method.
`TimeDiscreteImplicitNaturalGradientProjector`(pde)	A time-discrete natural gradient for solving temporal PDEs.
`TimeDiscreteNaturalGradientProjector`(pde, ...)	A time-discrete natural gradient projector for solving temporal PDEs.

class TimeDiscreteCollocationProjector(pde, **kwargs)[source]¶

Bases: CollocationProjector

Implement the Galerkin-based nonlinear projection method.

This subclass implements the assembly method to assemble the input and output tensors for a specific nonlinear projection problem using the Galerkin approach. It computes the approximation of a nonlinear problem by sampling collocation points and evaluating the corresponding function values.

Parameters:

pde (TemporalPDE | KineticPDE) – The PDE model to solve.
**kwargs – Additional parameters for the projection, including collocation points and losses.

sample_all_vars(**kwargs)[source]¶

Samples all variables required for the projection.

Include the collocation and boundary points.

Parameters:: **kwargs (Any) – Additional keyword arguments.
Return type:: tuple[LabelTensor, ...]
Returns:: A tuple containing sampled collocation points and boundary data.
Raises:: ValueError – If the approximation space type is not recognized.

assembly_post_sampling(data, **kwargs)[source]¶

Assembles the I/Otensors for the nonlinear Galerkin projection problem.

This method samples collocation points and evaluates the corresponding function values from the approximation space and the right-hand side of the problem. It then returns the tensors representing the inputs and outputs of the projection.

Parameters:

data (tuple[LabelTensor, ...]) – tuple containing sampled collocation points and boundary data.
**kwargs – Additional keyword arguments.

Return type:

tuple[tuple[Tensor, ...], tuple[Tensor, ...]]

Returns:

A tuple of tensors representing the inputs (u) and outputs (f) of the projection problem.

Raises:

ValueError – If the approximation space type is not recognized.

class TimeDiscreteNaturalGradientProjector(pde, **kwargs)[source]¶

Bases: TimeDiscreteCollocationProjector

A time-discrete natural gradient projector for solving temporal PDEs.

Parameters:

pde (TemporalPDE | KineticPDE) – The PDE model to solve.
**kwargs – Additional parameters for the projection, including collocation points and losses.

class TimeDiscreteImplicitNaturalGradientProjector(pde, bc_type='strong', **kwargs)[source]¶

Bases: TimeDiscreteCollocationProjector

A time-discrete natural gradient for solving temporal PDEs.

Parameters:

pde (TemporalPDE | KineticPDE) – The PDE model to solve.
bc_type (str) – The way the boundary condition is handled; “strong” for strongly, “weak” for weakly
**kwargs – Additional parameters for the projection, including collocation points and losses.

class TimeDiscreteAnagramProjector(pde, **kwargs)[source]¶

Bases: TimeDiscreteCollocationProjector

A time-discrete natural gradient projector for solving temporal PDEs.

Parameters:

pde (TemporalPDE | KineticPDE) – The PDE model to solve.
**kwargs – Additional parameters for the projection, including collocation points and losses.

class ExplicitTimeDiscreteScheme(pde, projector, projector_init=None, **kwargs)[source]¶

Bases: object

An explicit time-discrete scheme for solving a differential equation.

Use linear and/or nonlinear spaces.

The class supports initialization of the model with a target function, computation of the right-hand side (RHS) from the model, and stepping through time using a projector.

Parameters:

pde (TemporalPDE | KineticPDE) – The PDE model.
projector (TimeDiscreteCollocationProjector) – The projector for training the model.
projector_init (TimeDiscreteCollocationProjector | None) – The projector for initializing the model (if None, uses the same as projector).
**kwargs – Additional hyperparameters for the scheme.

initialization(**kwargs)[source]¶

Initializes the model by projecting the initial condition onto the model.

Parameters:: **kwargs (Any) – Additional parameters for the initialization, such as the number of epochs and collocation points.

abstract update(t, dt, **kwargs)[source]¶

Updates the model parameters using the time step and the chosen time scheme.

Parameters:

t (float) – The current time.
dt (float) – The time step.
**kwargs – Additional parameters for the update.

compute_relative_error_in_time(t, n_error=5000)[source]¶

Computes the relative error between the current and exact solution.

Parameters:

t (float) – The time at which the error is computed.
n_error (int) – The number of points used for computing the error. Default is 5_000.

Returns:

The L1, L2, and Linf errors.

Return type:

list

solve(dt=1e-05, final_time=0.1, sol_exact=None, **kwargs)[source]¶

Solves the full time-dependent problem, using time_step.

Parameters:

dt (float) – The time step.
final_time (float) – The final time.
sol_exact (Optional[Callable]) – The exact solution, if available.
**kwargs – Additional parameters for the time-stepping, such as the number of epochs, collocation points, and options for saving and plotting.