Paper:
We propose a novel approach to compute a stable and generalizing metric (LNSM) with convolutional neural networks (CNN) to compare field data from a variety of numerical simulation sources. Our method employs a Siamese network architecture that is motivated by the mathematical properties of a metric and is known to work well for finding similarities of other data modalities. We leverage a controllable data generation setup with partial differential equation (PDE) solvers to create increasingly different outputs from a reference simulation. In addition, the data generation allows for adjusting the difficulty of the resulting learning task. A central component of our learned metric is a specialized loss function, that introduces knowledge about the correlation between single data samples into the training process. To demonstrate that the proposed approach outperforms existing simple metrics for vector spaces and other learned, image based metrics we evaluate the different methods on a large range of test data. Additionally, we analyze generalization benefits of using the proposed correlation loss and the impact of an adjustable training data difficulty.

Review:
This is a well-written paper which looks into options for learning similarity metrics on data derived from PDE models common in the sciences. In comparison to other metric learning settings, here a type of ground-truth distance information is available (rather than, say, triplets), and it is possible to attempt to directly target an objective function which aims to match the learned distance to the ground-truth distance. The model architecture follows a fairly standard siamese-network setup.
Quite a bit of space is devoted to ensuring that the learned metric actually satisfies pseudo-metric axioms. This is all very clearly presented, with justifications for different modeling choices and how they preserve the axioms; my only criticism here is that many aspects of this are fairly obvious (i.e. an architecture which shares weights in computing the embeddings of both data points, followed by computing a squared L2 distance, will quite clearly get us in the ballpark of a pseudometric), but in my opinion "excess" clarity is much better than the opposite.
I think the more important contribution of the paper is in sections 4 and 5, which outlines a specific data generation process, including means of injecting noise, and compares options for loss functions. I would have expected pearson correlation to work quite well in this context, and it is interesting to note that performance notably improved by also adding an MSE term. I am curious about the "distance" prediction, as described just before equation 5, where it is stated that d \in R^n — is this really R^n? The target distance c is in [0,1]^n, and it seems like a simple modification to the distance prediction network would be capable of ensuring that the predicted values d also fall in this range. Such normalization could reduce the need for the MSE loss term, which presumably helps keep the overall relative scales of the two distances in check.
The empirical testing is also thorough, and I particularly appreciate the use of the random-weight networks as a baseline — I think it is good to note that these are actually fairly competitive on many of the test data sets (in fact, I believe it should be in bold for "TID" in table 1). 
I think the main weakness of this paper is that it falls slightly short of actually presenting the real use cases and needs for a similarity metric on PDE outputs — in my opinion, this comes to play when matching the output of a PDE-based model with real data. It would be nice to see a discussion of how this could be useful for parameter inference in PDE models. If there are other important applications of a distance learned in this way, I think the paper could benefit *greatly* by pointing them out. Otherwise, this risks being perceived as adding little value, since for individual PDE runs with known parameters, there is a ground-truth distance available — in which case, why bother using deep learning to estimate the distance, if the parameters are known? I think relevance to applications should to be clearly addressed.
The supplemental material is long, but complete and clearly presented.