Pointwise Detection of Well-Trained Regions in Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) rely on residual losses to enforce governing equations, but a small residual at an interior point does not guarantee that the predicted value matches the solution selected by the boundary or initial data. We study this pointwise detection problem: how to identify, during training, which domain points are already well trained using only quantities available to the PINN. We introduce a boundary-seeded detector that grows a well-trained set through local parent–child admissions. A candidate is admitted only when its own residual is small, its parent's residual remains small, and the parent's prediction has not drifted from the value stored at admission. The residual threshold is predicted from PDE-level features by a small structural meta-model, avoiding online estimation of stability constants or parameter-space sampling. The same primitive supports two uses: a conservative coverage-based stopping signal and an adaptive-domain training scheme with stored-value self-distillation. Across multiple PDE families, the detector reduces the false-stop rate from at least 40.9% for aggregate residual and residual–boundary criteria to 6.8%, while the detection-driven frontier improves training on a stiff convection benchmark and Helmholtz.