Decomposing the learned-versus-parametric advantage in elastodynamic damage inversion: estimation, representation, and data budget

Recovering an internal damage field from surface elastic-wave measurements is an ill-posed coefficient inverse problem, classically addressed by restricting the solution to a low-dimensional parametric family. Learned inversion removes that restriction and is known to outperform parametric methods in seismic velocity inversion; less often measured is why. We decompose the advantage by inserting a third reference between the two approaches: the best reconstruction the parametric class can produce when supplied with the true defect parameters. On a two-dimensional elastodynamic problem with distributed damage, we compare an encoder–decoder network trained on 8000 finite-difference simulations against direct parametric inversion, on the single-defect subset where the parametric method is strongest — given the true defect width and five restarts. Direct inversion reaches a relative L2 error of 0.435, the parametric ceiling 0.120, and the network 0.043: an estimation gap of 3.6 and a representation gap of 2.8. The ceiling is shown to be the true limit of the model class under four projection conventions, including the least-squares optimum. A nested learning curve locates each component. The estimation advantage is present at 500 samples with separated distributions; the representation advantage is unambiguous at 4000. The network’s error shows no detectable dependence on target anisotropy over the trained range, and degrades most on out-of-distribution anisotropic cracks, where IoU falls to 0.492 ± 0.009 over three training seeds. In forward-simulation count the learned approach becomes cheaper than direct inversion beyond 40 samples, per acquisition geometry. Retraining from scratch on a training distribution widened with the crack family recovers that family (IoU 0.492 → 0.790) at no measurable cost to the nominal regime, and an equal-size volume control attributes the gain to the composition of the training distribution rather than to its size. The study is entirely computational: the same solver generates and inverts the data, no measurement noise is modelled, and the input normalisation is numerically equivalent to subtracting a clean reference simulation. We also report a pre-registered prediction, on the behaviour of the train/validation gap with dataset size, that the measurements falsified.