Recovering a target signal from incomplete observations is a fundamental challenge in wireless communications, medical imaging and other fields. The process often depends on physical parameters that cannot be observed directly, such as channel state information and carrier-frequency offset in wireless links or coil sensitivity maps in parallel magnetic resonance imaging (MRI). Although these parameters are not the final targets, supervised parameter estimators normally require them as labels, which may be unreliable or unavailable. Researchers from the National University of Defense Technology and the Academy of Military Science have developed physics-embedded inverse learning (PEIL) to overcome this limitation. The study was published in National Science Review.
PEIL retains the familiar “estimate-then-solve” structure of physical inverse problems. A neural network predicts the parameters required by a physical solver, which then uses them to recover the target signal. The solver itself is not trained but remains differentiable. During training, the recovered signal is compared with a reference signal, and the error is propagated back through the solver to update the parameter estimator. The network can therefore learn to estimate intermediate parameters from the final recovery error, without being given parameter labels.
The team first tested PEIL in a high-mobility wireless system. From pilot observations, the network predicted carrier-frequency offset (CFO) and channel state information (CSI), while a fixed receiver used them to recover transmitted symbols. Training relied only on symbol-recovery error. Without CFO or CSI labels, PEIL learned to estimate physically meaningful communication parameters and achieved strong symbol recovery. At signal-to-noise ratios above 30 dB, PEIL achieved a lower symbol error rate than a least-squares baseline given the exact CFO and ground-truth channel responses at the pilot positions, with the same fixed interpolation. After training on one channel profile at 120 km/h, PEIL outperformed parameter-supervised baselines on unseen channel profiles and at speeds up to 360 km/h, without retraining. PEIL reached comparable recovery performance using only one twentieth as many training samples as the parameter-supervised baseline.
The experiments further showed that more accurate estimates of individual parameters do not always lead to better signal recovery. Within a limited range, a small CFO deviation can be offset by a corresponding adjustment in CSI phase, allowing different parameter combinations to recover nearly identical symbols. PEIL learned coordinated CFO and CSI estimates that jointly supported signal recovery by the fixed receiver. The researchers describe these as “task-optimal” parameters because they are selected according to final recovery performance. This does not mean that inaccurate parameters are generally preferable; when the parameters themselves are the quantities of interest, their physical accuracy remains essential.
To test whether this principle extends beyond wireless communications, the team applied PEIL to parallel MRI. The network predicted coil sensitivity maps from undersampled multi-coil data, and a fixed physical solver used them to reconstruct images. Without sensitivity-map labels or a separate calibration step, PEIL estimated stable coil sensitivity maps and reconstructed anatomically faithful images under fourfold and eightfold undersampling. In the MRI implementation, the network also provided an image-domain reference to the fixed solver. Relative to this reference, the final physics-based reconstruction improved peak signal-to-noise ratio by 5.5 to 6.1 dB.
Across wireless communications and MRI, PEIL changes the basis of parameter supervision: learning is guided by final task performance rather than ground-truth labels for intermediate parameters. When such parameters are used by a differentiable physical solver and the resulting output can be evaluated, “whether the result is correct” can itself provide supervision for parameter estimation. This principle may extend to other inverse problems in sensing, communications and computational imaging.
National Science Review
Computational simulation/modeling