Causal Physics Informed Single Model Solver#

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

class CausalPhysicsInformedSingleModelSolver(problem, model, optimizer=None, scheduler=None, weighting=None, loss=None, eps=100, n_steps=10, regularized_conditions=None)[source]

Bases: PhysicsInformedMixin, SingleModelSolver

Single-model solver for causal physics-informed learning problems.

This solver approximates the solution of a time-dependent differential problem using a single model and a causality-aware training objective. It is intended for problems whose conditions include equation residuals and boundary residuals evaluated across ordered time snapshots. It can be used only for forward problems, due to the causal nature of the training objective.

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

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

The solver minimizes a causal residual loss that weights each time snapshot according to the residuals accumulated at previous times. For a time dependent 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

\[\mathcal{L}_{\mathrm{problem}} = \frac{1}{N_t} \sum_{i=1}^{N_t} \omega_i \mathcal{L}_r(t_i),\]

where the residual loss at time \(t\) is

\[\mathcal{L}_r(t) = \frac{1}{N_{\Omega}} \sum_{j=1}^{N_{\Omega}} \mathcal{L}\left( \mathcal{A}[\hat{\mathbf{u}}](\mathbf{x}_j, t) \right) + \frac{1}{N_{\partial\Omega}} \sum_{j=1}^{N_{\partial\Omega}} \mathcal{L} \left( \mathcal{B}[\hat{\mathbf{u}}](\mathbf{x}_j, t) \right).\]

The causal weights are defined as

\[\omega_i = \exp \left( -\epsilon \sum_{k=1}^{i-1} \mathcal{L}_r(t_k) \right),\]

where \(\epsilon\) is a hyperparameter controlling the strength of the causal weighting, and \(\mathcal{L}\) is the selected loss function, typically the mean squared error.