Gradient Physics Informed Single Model Solver#

Module for the gradient physics-informed single-model solver class.

class GradientPhysicsInformedSingleModelSolver(problem, model, optimizer=None, scheduler=None, weighting=None, loss=None, regularization_weight=1.0, regularized_conditions=None)[source]

Bases: PhysicsInformedMixin, GradientEnhancedMixin, SingleModelSolver

Single-model solver for gradient-enhanced physics-informed learning problems.

This solver approximates the solution of a differential problem using a single model and augments the standard physics-informed objective with gradient-enhanced residual terms. It can be used for both forward and inverse problems.

Given a model \(\mathcal{M}\), the predicted solution is

\[\hat{\mathbf{u}}(\mathbf{x}) = \mathcal{M}(\mathbf{x}).\]

The solver minimizes both the residuals of the differential operators defining the problem and the gradients of those residuals with respect to the input variables. For a problem with governing equation operator \(\mathcal{A}\) in the domain \(\Omega\) and boundary operator \(\mathcal{B}\) on the boundary \(\partial\Omega\), the objective can be written as

\[\mathcal{L}_{\mathrm{problem}} = \frac{1}{N_{\Omega}} \sum_{i=1}^{N_{\Omega}} \mathcal{L} \left( \mathcal{A}[\hat{\mathbf{u}}](\mathbf{x}_i) \right) + \frac{1}{N_{\partial\Omega}} \sum_{i=1}^{N_{\partial\Omega}} \mathcal{L} \left( \mathcal{B}[\hat{\mathbf{u}}](\mathbf{x}_i) \right) + \frac{1}{N_{\Omega}} \sum_{i=1}^{N_{\Omega}} \mathcal{L} \left( \nabla_{\mathbf{x}} \mathcal{A}[\hat{\mathbf{u}}](\mathbf{x}_i) \right) + \frac{1}{N_{\partial\Omega}} \sum_{i=1}^{N_{\partial\Omega}} \mathcal{L} \left( \nabla_{\mathbf{x}} \mathcal{B}[\hat{\mathbf{u}}] (\mathbf{x}_i) \right),\]

where \(\mathcal{L}\) is the selected loss function, typically the mean squared error.