GradientPINN#

class GradientPINN(problem, model, optimizer=None, scheduler=None, weighting=None, loss=None)[source]#

Bases: PINN

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

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

minimizing the loss function;

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

where \(\mathcal{L}\) is a specific loss function, typically the MSE:

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

GradientPINN#

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