PINN#

class PINN(problem, model, extra_features=None, loss=MSELoss(), optimizer=<class 'torch.optim.adam.Adam'>, optimizer_kwargs={'lr': 0.001}, scheduler=<class 'torch.optim.lr_scheduler.ConstantLR'>, scheduler_kwargs={'factor': 1, 'total_iters': 0})[source]#

Bases: PINNInterface

Physics Informed Neural Network (PINN) solver class. This class implements Physics Informed Neural Network solvers, using a user specified model to solve a specific problem. It can be used for solving both forward and inverse problems.

The Physics Informed Network aims to find the solution \(\mathbf{u}:\Omega\rightarrow\mathbb{R}^m\) of the differential problem:

\[\begin{split}\begin{cases} \mathcal{A}[\mathbf{u}](\mathbf{x})=0\quad,\mathbf{x}\in\Omega\\ \mathcal{B}[\mathbf{u}](\mathbf{x})=0\quad, \mathbf{x}\in\partial\Omega \end{cases}\end{split}\]

minimizing the loss function

\[\mathcal{L}_{\rm{problem}} = \frac{1}{N}\sum_{i=1}^N \mathcal{L}(\mathcal{A}[\mathbf{u}](\mathbf{x}_i)) + \frac{1}{N}\sum_{i=1}^N \mathcal{L}(\mathcal{B}[\mathbf{u}](\mathbf{x}_i))\]

where \(\mathcal{L}\) is a specific loss function, default Mean Square Error:

\[\mathcal{L}(v) = \| v \|^2_2.\]

See also

Original reference: Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3, 422-440. DOI: 10.1038.

Parameters:

  • problem (AbstractProblem) – The formulation of the problem.

  • model (torch.nn.Module) – The neural network model to use.

  • loss (torch.nn.Module) – The loss function used as minimizer, default torch.nn.MSELoss.

  • extra_features (torch.nn.Module) – The additional input features to use as augmented input.

  • optimizer (torch.optim.Optimizer) – The neural network optimizer to use; default is torch.optim.Adam.

  • optimizer_kwargs (dict) – Optimizer constructor keyword args.

  • scheduler (torch.optim.LRScheduler) – Learning rate scheduler.

  • scheduler_kwargs (dict) – LR scheduler constructor keyword args.

forward(x)[source]#

Forward pass implementation for the PINN solver. It returns the function evaluation \(\mathbf{u}(\mathbf{x})\) at the control points \(\mathbf{x}\).

Parameters:

x (LabelTensor) – Input tensor for the PINN solver. It expects a tensor \(N \times D\), where \(N\) the number of points in the mesh, \(D\) the dimension of the problem,

Returns:

PINN solution evaluated at contro points.

Return type:

LabelTensor

loss_phys(samples, equation)[source]#

Computes the physics loss for the PINN solver based on given samples and equation.

Parameters:

  • samples (LabelTensor) – The samples to evaluate the physics loss.

  • equation (EquationInterface) – The governing equation representing the physics.

Returns:

The physics loss calculated based on given samples and equation.

Return type:

LabelTensor

configure_optimizers()[source]#

Optimizer configuration for the PINN solver.

Returns:

The optimizers and the schedulers

Return type:

tuple(list, list)

property scheduler#

Scheduler for the PINN training.

property neural_net#

Neural network for the PINN training.