scimba_torch.numerical_solvers.pinn_preconditioners

Implementations of PINN preconditioners.

class AnagramPreconditioner(space, pde, **kwargs)[source]

Bases: MatrixPreconditionerPinn

Anagram preconditioner.

This preconditioner is based on the anagram method, which aims to improve convergence by transforming the problem into a more favorable form.

Parameters:
Keyword Arguments:

svd_threshold (float) – Threshold for singular value decomposition (default: 1e-6).

compute_preconditioning_matrix(labels, *args, **kwargs)[source]

Compute the preconditioning matrix using the main operator.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The preconditioning matrix.

compute_preconditioning_matrix_bc(labels, *args, **kwargs)[source]

Compute the boundary condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The boundary condition preconditioning matrix.

compute_preconditioning_matrix_ic(labels, *args, **kwargs)[source]

Compute the initial condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The initial condition preconditioning matrix.

compute_preconditioning_matrix_dl(*args, **kwargs)[source]

Computes the Gram matrix of the network for the given input tensors.

Parameters:

  • *args (Tensor) – Input tensors for computing the Gram matrix.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The computed Gram matrix.

assemble_right_member(data, res_l, res_r)[source]

Assemble the right-hand side of the equation.

Parameters:

  • data (tuple | dict) – Input data, either as a tuple or a dictionary.

  • res_l (tuple) – Left residuals.

  • res_r (tuple) – Right residuals.

Return type:

Tensor

Returns:

The assembled right-hand side tensor.

assemble_right_member_bc(data, res_l, res_r)[source]

Assemble the right-hand side for boundary conditions.

Parameters:

  • data (tuple | dict) – Input data, either as a tuple or a dictionary.

  • res_l (tuple) – Left residuals.

  • res_r (tuple) – Right residuals.

Return type:

Tensor

Returns:

The assembled right-hand side tensor for boundary conditions.

assemble_right_member_ic(data, res_l, res_r)[source]

Assemble the right-hand side for initial conditions.

Parameters:

  • data (tuple | dict) – Input data, either as a tuple or a dictionary.

  • res_l (tuple) – Left residuals.

  • res_r (tuple) – Right residuals.

Return type:

Tensor

Returns:

The assembled right-hand side tensor for initial conditions.

assemble_right_member_dl(data, res_l, res_r)[source]

Assembles the right member for the preconditioning.

Parameters:

  • data (tuple | dict) – The input data for assembling the right member.

  • res_l (tuple) – The left residuals.

  • res_r (tuple) – The right residuals.

Return type:

Tensor

Returns:

The assembled right member.

class EnergyNaturalGradientPreconditioner(space, pde, **kwargs)[source]

Bases: MatrixPreconditionerPinn

Energy-based natural gradient preconditioner.

Parameters:
Keyword Arguments:

  • matrix_regularization (float) – Regularization parameter for the preconditioning matrix (default: 1e-6).

  • adaptive_matrix_regularization (bool, default=False)

  • training (If True, adaptively adjusts the regularization parameter during)

  • Levenberg-Marquardt). ((à la)

  • parameter (If False, uses the fixed regularization)

  • matrix_regularization. (specified by)

  • adaptive_matrix_regularization_increase (float, default=10.0)

  • search (Factor by which to decrease the regularization parameter if the line)

  • size. (quickly succeeds in finding a suitable step)

  • adaptive_matrix_regularization_decrease (float, default=0.5)

  • search

  • size.

  • adaptive_matrix_regularization_min (float, default=1e-12)

  • adjustment. (Minimum value for the regularization parameter when using adaptive)

  • use_lstsq (bool) – Whether to use least squares solver for

  • (default (computing the preconditioner) – True).

  • inversion. (If False, uses direct matrix)

compute_preconditioning_matrix(labels, *args, **kwargs)[source]

Compute the preconditioning matrix using the main operator.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The preconditioning matrix.

compute_preconditioning_matrix_bc(labels, *args, **kwargs)[source]

Compute the boundary condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The boundary condition preconditioning matrix.

compute_preconditioning_matrix_ic(labels, *args, **kwargs)[source]

Compute the initial condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The initial condition preconditioning matrix.

compute_preconditioning_matrix_dl(*args, **kwargs)[source]

Computes the Gram matrix of the network for the given input tensors.

Parameters:

  • *args (Tensor) – Input tensors for computing the Gram matrix.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The computed Gram matrix.

compute_full_preconditioning_matrix(data, **kwargs)[source]

Compute the full preconditioning matrix.

Include contributions from the main operator, boundary conditions, and initial conditions.

Parameters:

  • data (tuple | dict) – Input data, either as a tuple or a dictionary.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The full preconditioning matrix.

update_matrix_regularization(n_steps, loss_has_decreased)[source]

Updates the regularization parameter for the preconditioning matrix.

This method adaptively adjusts the regularization parameter during training based on the number of line search steps taken and whether the loss has decreased.

Parameters:

  • n_steps (int) – The number of line search steps taken.

  • loss_has_decreased (bool) – Whether the loss has decreased after the line search.

reset_matrix_regularization()[source]

Resets the regularization parameter to its initial value.

class NystromNaturalGradientPreconditioner(space, pde, **kwargs)[source]

Bases: MatrixPreconditionerPinn

Randomized matrix-free natural gradient preconditioner.

Implements [Bioli and al. 2025]

Parameters:
Keyword Arguments:

  • matrix_free (bool) – Use jvp and vjp to compute the matrix vector product function instead of Phi matrix. (default: False).

  • eps (float) – The eps for adaptive matrix regularization. (default: torch.finfo(torch.get_default_dtype()).eps).

  • debug (bool) – Debugging mode - check that gram matrices are coherent. (default: False).

Raises:

NotImplementedError – data loss and matrix free option not implemented

[Bioli and al. 2025] Accelerating Natural Gradient Descent for PINNs with

Randomized Nyström Preconditioning, I. Bioli, C. Marcati, G. Sangalli https://arxiv.org/abs/2505.11638v3

compute_preconditioning_matrix(labels, *args, **kwargs)[source]

Compute the preconditioning matrix using the main operator.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The preconditioning matrix.

compute_preconditioning_matrix_bc(labels, *args, **kwargs)[source]

Compute the boundary condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The boundary condition preconditioning matrix.

compute_preconditioning_matrix_ic(labels, *args, **kwargs)[source]

Compute the initial condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The initial condition preconditioning matrix.

compute_preconditioning_matrix_dl(*args, **kwargs)[source]

Computes the Gram matrix of the network for the given input tensors.

Parameters:

  • *args (Tensor) – Input tensors for computing the Gram matrix.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The computed Gram matrix.

class SketchyNaturalGradientPreconditioner(space, pde, **kwargs)[source]

Bases: MatrixPreconditionerPinn

Sketchy natural gradient preconditioner.

Implements [McKay and al. 2025].

Parameters:
Keyword Arguments:

  • tol (float) – The threshold for deciding the size of the sketch matrix, r in the paper. (default: 1e-13).

  • single_pass (bool) – single pass or two pass (default False, means two pass).

[McKay and al. 2025] Near-optimal Sketchy Natural Gradients

for Physics-Informed Neural Networks, M. B. McKay, A. Kaur, C. Greif, B. Wetton Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025

compute_preconditioning_matrix(labels, *args, **kwargs)[source]

Compute the preconditioning matrix using the main operator.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The preconditioning matrix.

compute_preconditioning_matrix_bc(labels, *args, **kwargs)[source]

Compute the boundary condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The boundary condition preconditioning matrix.

compute_preconditioning_matrix_ic(labels, *args, **kwargs)[source]

Compute the initial condition preconditioning matrix.

Parameters:

  • labels (Tensor) – The labels tensor.

  • *args (Tensor) – Additional arguments.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The initial condition preconditioning matrix.

compute_preconditioning_matrix_dl(*args, **kwargs)[source]

Computes the Gram matrix of the network for the given input tensors.

Parameters:

  • *args (Tensor) – Input tensors for computing the Gram matrix.

  • **kwargs – Additional keyword arguments.

Return type:

Tensor

Returns:

The computed Gram matrix.

Modules

anagram_ng

Preconditioner for pinns.

energy_ng

Preconditioner for pinns.

nystrom_ng

Preconditioner for pinns.

sketchy_ng

Preconditioner for pinns.