Discovery of unobservable parameters via physical embedding.

Recovering a source signal from indirect measurements often depends on parameters that cannot be observed directly, such as wireless channel states or MRI coil sensitivities. We introduce Physics-Embedded Inverse Learning (PEIL), which embeds a fixed, differentiable inverse solver and uses signal recovery to supervise their estimation without parameter labels. PEIL makes parameter selection task-optimal: each setting is judged by the reconstruction it produces through the fixed solver, rather than by its agreement with nominal labels. In locally non-identifiable regimes, different settings yield nearly equivalent reconstructions, allowing the loss to favour those better suited to the fixed solver. Non-identifiability thus becomes a resource for recovery. In high-mobility wireless communications, PEIL generalises to unseen channel profiles and velocities without retraining and reaches comparable recovery performance using 20-fold fewer training samples than baselines trained with parameter labels. At high SNR, it also achieves a lower symbol error rate than an oracle-reference least-squares baseline using true pilot-channel values and fixed interpolation. In parallel MRI, without sensitivity-map labels or a separate calibration step, PEIL discovers stable coil sensitivity maps with magnitude structure consistent with a calibration reference and reconstructs anatomically faithful images. Together, these results show that a fixed inverse solver can make unobservable parameters learnable from signal recovery.