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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.<\/jats:p>","DOI":"10.1093\/nsr\/nwag508","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T11:38:57Z","timestamp":1787053137000},"source":"Crossref","is-referenced-by-count":1,"title":["Discovery of unobservable parameters via physical embedding"],"prefix":"10.1093","volume":"13","author":[{"given":"Le","family":"Cheng","sequence":"first","affiliation":[{"name":"College of Electronic Science and Technology, National University of Defense Technology , Changsha 410073 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoran","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Electronic Science and Technology, National University of Defense Technology , Changsha 410073 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