scimba_torch.approximation_space.kernelx_space

Defines the Kernel-based approximation space and its components.

Classes

ExponentialKernel(**kwargs)	Exponential kernel approximation space.
GaussianKernel(**kwargs)	Gaussian kernen approximation space.
KernelxSpace(nb_unknowns, nb_parameters, ...)	A nonlinear approximation space using a neural network model.
MultiquadraticKernel(**kwargs)	Multiquadratic kernel approximation space.

class GaussianKernel(**kwargs)

Bases: Module

Gaussian kernen approximation space.

Parameters:

**kwargs (Any) – Additional arguments for the Gaussian kernel.

forward(vector_diff, aniso)

Compute the Gaussian kernel values.

Parameters:

• vector_diff (Tensor) – Tensor of shape (num_samples, num_centers, input_dim) representing the difference between input points and kernel centers.

• aniso (Tensor) – Tensor of shape (num_centers, input_dim, input_dim) representing the anisotropic transformation matrices for each kernel center.

Return type:

Tensor

Returns:

Tensor of shape (num_samples, num_centers) containing the Gaussian kernel

values.

class ExponentialKernel(**kwargs)

Bases: Module

Exponential kernel approximation space.

Parameters:

**kwargs (Any) –

Additional arguments for the Exponential kernel.

• beta: Shape parameter of the Exponential kernel (default: 2).

forward(vector_diff, aniso)

Compute the Exponential kernel values.

Parameters:

• vector_diff (Tensor) – Tensor of shape (num_samples, num_centers, input_dim) representing the difference between input points and kernel centers.

• aniso (Tensor) – Tensor of shape (num_centers, input_dim, input_dim) representing the anisotropic transformation matrices for each kernel center.

Return type:

Tensor

Returns:

Tensor of shape (num_samples, num_centers) containing the Exponential

kernel values.

class MultiquadraticKernel(**kwargs)

Bases: Module

Multiquadratic kernel approximation space.

Parameters:

**kwargs (Any) –

Additional arguments for the Multiquadratic kernel.

• beta: Shape parameter of the Multiquadratic kernel (default: 2).

forward(vector_diff, aniso)

Compute the Multiquadratic kernel values.

Parameters:

• vector_diff (Tensor) – Tensor of shape (num_samples, num_centers, input_dim) representing the difference between input points and kernel centers.

• aniso (Tensor) – Tensor of shape (num_centers, input_dim, input_dim) representing the anisotropic transformation matrices for each kernel center.

Return type:

Tensor

Returns:

Tensor of shape (num_samples, num_centers) containing the Multiquadratic

kernel values.

class KernelxSpace(nb_unknowns, nb_parameters, kernel_type, nb_centers, spatial_domain, integrator, **kwargs)

Bases: AbstractApproxSpace, ScimbaModule

A nonlinear approximation space using a neural network model.

This class represents a parametric approximation space, where the solution is modeled by a neural network. It integrates functionality for evaluating the network, setting/retrieving degrees of freedom, and computing the Jacobian.

Parameters:

• nb_unknowns (int) – Number of unknowns in the approximation problem.

• nb_parameters (int) – Number of parameters in the approximation problem.

• kernel_type (Module) – The type of kernel to use for the approximation.

• nb_centers (int) – Number of centers for the kernel functions.

• spatial_domain (VolumetricDomain) – The spatial domain of the problem.

• integrator (TensorizedSampler) – Sampler used for integration over the spatial and parameter domains.

• **kwargs – Additional arguments passed to the neural network model.

centers: torch.nn.Parameter

Centers of the kernel functions, initialized as learnable parameters.

beta: float

Parameter for the kernel functions, controlling their shape and behavior.

M_aniso: torch.nn.Parameter

Anisotropic transformation matrix for the kernel functions, initialized as learnable parameters.

eps: torch.nn.Parameter

Epsilon parameters for the kernel functions, initialized as learnable parameters.

ndof: int

Total number of degrees of freedom in the network.

forward(features, with_last_layer=True)

Forward pass through the kernel model.

Parameters:

• features (Tensor) – Input tensor with concatenated spatial and parameter data.

• with_last_layer (bool) – Whether to apply the final linear layer.

Return type:

Tensor

Returns:

Output tensor from the kernel model.

evaluate(x, mu, with_last_layer=True)

Evaluate the parametric model for given inputs and parameters.

Parameters:

• x (LabelTensor) – Input tensor from the spatial domain.

• mu (LabelTensor) – Input tensor from the parameter domain.

• with_last_layer (bool) – Whether to apply the final linear layer.

Return type:

MultiLabelTensor

Returns:

Output tensor from the neural network, wrapped with multi-label metadata.

set_dof(theta, flag_scope='all')

Sets the degrees of freedom (DoF) for the neural network.

Parameters:

• theta (Tensor) – A vector containing the network parameters.

• flag_scope (str) – The scope of parameters to return.

Return type:

None

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

Retrieves the degrees of freedom (DoF) of the neural network.

Parameters:

• flag_scope (str) – The scope of parameters to return.

• flag_format (str) – The format of the returned parameters.

Return type:

Tensor

Returns:

Tensor containing the DoF of the network.

jacobian(x, mu)

Compute the Jacobian of the network with respect to its parameters.

Parameters:

• x (LabelTensor) – Input tensor from the spatial domain.

• mu (LabelTensor) – Input tensor from the parameter domain.

Return type:

Tensor

Returns:

Jacobian matrix of shape (num_samples, out_size, num_params).