SelfAdaptivePINN#

class SelfAdaptivePINN(problem, model, weight_function=Sigmoid(), optimizer_model=None, optimizer_weights=None, scheduler_model=None, scheduler_weights=None, weighting=None, loss=None)[source]#

Bases: PINNInterface, MultiSolverInterface

Self-Adaptive Physics-Informed Neural Network (SelfAdaptivePINN) solver class. This class implements the Self-Adaptive Physics-Informed Neural Network solver, using a user specified model to solve a specific problem. It can be used to solve both forward and inverse problems.

The Self-Adapive Physics-Informed Neural Network solver aims to find the solution \(\mathbf{u}:\Omega\rightarrow\mathbb{R}^m\) of a 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 pointwise loss evaluation using a mask :math:m and self-adaptive weights, which allow the model to focus on regions of the domain where the residual is higher.

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.

The Self-Adaptive Physics-Informed Neural Network solver identifies the solution and appropriate self adaptive weights by solving the following optimization 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, , typically the MSE:

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

SelfAdaptivePINN#

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