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This manuscript presents an in-depth case study of glacier ice thickness inversion for the Bhote Kosi glacier in the Himalaya, which is performed using the deep-learning-based Instructed Glacier Model (IGM). The authors invert the observed surface velocities for the ice thickness, and the L-curve technique is utilized as to selecting the regularization parameter. The results are validated against the recent airborne radar thickness observations of Pritchard et al. (2026). Furthermore, cross-validation experiments with thickness-constrained inversions have been performed, and it is found that the thickness observations correct the modelled field locally, especially in the low-velocity areas, however they have a weak interpolative power at the unconstrained locations. The study is concluded with practical recommendations as to the choice of the inversion parameters and the placement of the future thickness measurements.

• It is noteworthy that the manuscript is written very well.

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  The introduction has a very good flow and brings even the readers who are not so close to the mountain glacier research up to speed with the topic.

• Line 144: it is stated that the approach differs from PINNs in two ways, however these differences are not explained clearly. A bit more clarification would be of help here.

• It is understood that the model is based on the work of Jouvet and Cordonnier (2023), however the reader should be able to fully understand the approach and the method without having read that publication. A more standalone explanation of the relevant details from that work would be appreciated.

• It is not clear which of the two following aspects makes the model a PINN: (1) the fact that the weights λ of the CNN are trained to minimize the energy functional associated with the Blatter-Pattyn model, or (2) the fact that the aim is to obtain an optimal fit of the control variables across the whole glacier, in such a way that the misfit between the surface velocity output from the emulator (Eq. 1) and the observed surface velocity is small (Jouvet, 2023). Some clarification on this point would be of use.

• Line 191: it is not clear why the pre-trained model undergoes only one iteration of retraining and not more. A brief justification would be useful.

• The authors have shown that the lack of thickness observations need not be a barrier to producing a reasonable thickness estimate, which is a very important result and is nicely applicable to the other glaciers around the world. However, it is also acknowledged that the method is unable to produce similar results for the glaciers with no measured thickness. Demonstrating this limitation with an example would provide a valuable proof of the limitations of the approach and would add to the completeness of the paper.

• The importance which is given to advising on the observation locations for the low-velocity glaciers is very significant and addresses a gap in the available observed data.

• The inversion is initialized with the thickness estimate of Millan et al. (2022) (Sect. 2.2.1), while the observed velocities also come from the same study. In Sect. 4.3, the shared velocity data is acknowledged, however the shared initialization is not: due to the fact that the cost function is non-convex, the final thickness field might retain some dependence on the initial guess. A sensitivity test with a different initialization (e.g. the consensus estimate, or a uniform thickness) would confirm that the result is independent of the starting point and would strengthen the comparison in Fig. 6.

• There is a temporal mismatch between the input datasets: the DEM is from 2010–2015, the velocities are from 2017–2018, and the thickness observations are from late 2019. It is mentioned in the conclusions that the contemporaneity between the input datasets is important, however this is not assessed for Bhote Kosi itself. The surface lowering between the DEM epoch and the radar survey could bias the comparison with a positive expected sign, which is likely still very small. A brief statement of the expected sign and magnitude of this effect would be valuable.