GPINN#

class GPINN(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: PINN

Gradient Physics Informed Neural Network (GPINN) solver class. This class implements Gradient 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 Gradient 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

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

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

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