SAPINN#

class SAPINN(problem, model, weights_function=Sigmoid(), extra_features=None, loss=MSELoss(), optimizer_model=<class 'torch.optim.adam.Adam'>, optimizer_model_kwargs={'lr': 0.001}, optimizer_weights=<class 'torch.optim.adam.Adam'>, optimizer_weights_kwargs={'lr': 0.001}, scheduler_model=<class 'torch.optim.lr_scheduler.ConstantLR'>, scheduler_model_kwargs={'factor': 1, 'total_iters': 0}, scheduler_weights=<class 'torch.optim.lr_scheduler.ConstantLR'>, scheduler_weights_kwargs={'factor': 1, 'total_iters': 0})[source]#

Bases: PINNInterface

Self Adaptive Physics Informed Neural Network (SAPINN) solver class. This class implements Self-Adaptive 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 Self Adapive Physics Informed Neural Network aims to find the solution \(\mathbf{u}:\Omega\rightarrow\mathbb{R}^m\) of the differential problem:

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

integrating the pointwise loss evaluation through a mask \(m\) and self adaptive weights that permit to focus the loss function on specific training samples. The loss function to solve the problem is

\[\mathcal{L}_{\rm{problem}} = \frac{1}{N} \sum_{i=1}^{N_\Omega} m \left( \lambda_{\Omega}^{i} \right) \mathcal{L} \left( \mathcal{A} [\mathbf{u}](\mathbf{x}) \right) + \frac{1}{N} \sum_{i=1}^{N_{\partial\Omega}} m \left( \lambda_{\partial\Omega}^{i} \right) \mathcal{L} \left( \mathcal{B}[\mathbf{u}](\mathbf{x}) \right),\]

denoting the self adaptive weights as \(\lambda_{\Omega}^1, \dots, \lambda_{\Omega}^{N_\Omega}\) and \(\lambda_{\partial \Omega}^1, \dots, \lambda_{\Omega}^{N_\partial \Omega}\) for \(\Omega\) and \(\partial \Omega\), respectively.

Self Adaptive Physics Informed Neural Network identifies the solution and appropriate self adaptive weights by solving the following problem

\[\min_{w} \max_{\lambda_{\Omega}^k, \lambda_{\partial \Omega}^s} \mathcal{L} ,\]

where \(w\) denotes the network parameters, and \(\mathcal{L}\) is a specific loss function, default Mean Square Error:

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