Competitive Physics-Informed Solver#

Module for the competitive physics-informed multi-model solver.

class CompetitivePhysicsInformedSolver(problem, model, discriminator=None, optimizer_model=None, optimizer_discriminator=None, scheduler_model=None, scheduler_discriminator=None, weighting=None, loss=None)[source]#

Bases: PhysicsInformedMixin, MultiModelSolver

Multi-model solver for competitive physics-informed learning problems.

This solver approximates the solution of a differential problem using a trainable model together with a discriminator network. It is intended for problems whose conditions may include supervised data, equation residuals evaluated on input points, and equation residuals sampled from domains.

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

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

The discriminator \(D\) assigns pointwise weights to the residuals, encouraging the model to focus on regions where the approximation performs poorly. The model parameters are optimized by minimizing the loss, while the discriminator parameters are optimized by maximizing it.

For a problem with governing equation operator \(\mathcal{A}\) in the domain \(\Omega\) and boundary operator \(\mathcal{B}\) on the boundary \(\partial\Omega\), the competitive objective can be written as

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

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

The model and discriminator are trained through a min-max problem:

\[\min_{\theta} \max_{\phi} \mathcal{L}_{\mathrm{problem}},\]

where \(\theta\) denotes the model parameters and \(\phi\) denotes the discriminator parameters.