Self-Adaptive Physics-Informed Solver#

Module for the self-adaptive physics-informed multi-model solver.

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

Bases: PhysicsInformedMixin, MultiModelSolver

Multi-model solver for self-adaptive physics-informed learning problems.

This solver approximates the solution of a differential problem using a trainable model together with condition-wise self-adaptive weights. 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

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

For each condition, the solver introduces trainable pointwise weights. These weights are passed through a user-defined weight function \(m\), typically chosen to keep the effective weights bounded or positive. The resulting weighted objective encourages the model to focus on regions where the residual is larger.

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

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

where \(\lambda_{\Omega}^{i}\) and \(\lambda_{\partial\Omega}^{i}\) are the self-adaptive weights associated with points in \(\Omega\) and \(\partial\Omega\), respectively, and \(\mathcal{L}\) is the selected loss function, typically the mean squared error.

The model parameters and the self-adaptive weights are optimized through a min-max problem:

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

where \(\theta\) denotes the model parameters and \(\lambda\) denotes the collection of self-adaptive weights.