scimba_torch.numerical_solvers.elliptic_pde.deep_ritz

Deep Ritz method for solving elliptic PDEs using deep learning.

Classes

DeepRitzElliptic(pde[, bc_type])	A deep learning-based solver for elliptic PDEs using the Ritz method.
NaturalGradientDeepRitzElliptic(pde[, bc_type])	A Deep Ritz solver for elliptic PDEs using natural gradient descent.

class DeepRitzElliptic(pde, bc_type='strong', **kwargs)

Bases: CollocationProjector

A deep learning-based solver for elliptic PDEs using the Ritz method.

The class inherits from AbstractNonlinearProjector and provides methods for assembling the PDE system and computing the metric matrix.

Parameters:

• pde (RitzFormEllipticPDE | DivAGradUPDE) – The elliptic PDE to be solved, represented as an instance of RitzForm_EllipticPDE.

• bc_type (str) – The type of boundary condition to be applied (“strong” or “weak”). (default: “strong”)

• **kwargs – Additional keyword arguments for losses and optimizers.

get_dof(flag_scope='all', flag_format='list')

Gets the degrees of freedom for the solver, including those from the PDE.

Parameters:

• flag_scope (str) – Scope flag for getting degrees of freedom. (default: “all”)

• flag_format (str) – Format flag for the degrees of freedom (“list” or “tensor”). (default: “list”)

Returns:

The degrees of freedom tensor or list.

evaluate(x, mu)

Evaluates the solution at given spatial and parameter points.

Parameters:

• x (Tensor) – Spatial points where the solution is evaluated.

• mu (Tensor) – Parameter points where the solution is evaluated.

Return type:

MultiLabelTensor

Returns:

The evaluated solution as a MultiLabelTensor.

metric_matrix(**kwargs)

Placeholder method for computing the metric matrix.

Parameters:

**kwargs (Any) – Additional keyword arguments.

Returns:

The metric matrix.

Return type:

torch.Tensor

Raises:

NotImplementedError – This method must be implemented in subclasses.

sample_all_vars(**kwargs)

Samples all necessary variables for assembling the PDE system.

Parameters:

**kwargs (Any) –

Additional keyword arguments including:

• n_collocation: Number of collocation points for the PDE. Defaults to 1000.

• n_bc_collocation: Number of collocation points for the boundary conditions. Defaults to 1000.

Return type:

tuple

Returns:

A tuple containing sampled spatial points, parameter points, and, if applicable, boundary points and normals.

assembly_post_sampling(xmu, **kwargs)

Assembles the PDE system after sampling all necessary variables.

Parameters:

• xmu (tuple) – A tuple containing sampled spatial points, parameter points, and, if applicable, boundary points and normals.

• **kwargs – Additional keyword arguments.

Return type:

tuple

Returns:

A tuple containing the assembled system of equations (Lo, f).

assembly(**kwargs)

Assembles the system of equations for the PDE.

(and weak boundary conditions if needed).

Parameters:

**kwargs (Any) –

Additional keyword arguments including:

• n_collocation (int): Number of collocation points for the PDE. Defaults to 1000.

• n_bc_collocation (int): Number of collocation points for the boundary conditions. Defaults to 1000.

Return type:

tuple

Returns:

A tuple containing the assembled system of equations (Lo, f).

class NaturalGradientDeepRitzElliptic(pde, bc_type='strong', **kwargs)

Bases: DeepRitzElliptic

A Deep Ritz solver for elliptic PDEs using natural gradient descent.

Parameters:

• pde (Laplacian2DDirichletRitzForm | DivAGradUPDE) – The elliptic PDE to be solved, represented as an instance of RitzForm_EllipticPDE.

• bc_type (str) – The type of boundary condition to be applied (“strong” or “weak”). (default: “strong”)

• **kwargs – Additional keyword arguments for losses, optimizers, and preconditioners.