  We propose a neural network-based approach that computes a stable and
generalizing metric (LSiM) to compare data from a variety of numerical
simulation sources. We focus on scalar time-dependent 2D data that commonly
arises from motion and transport-based partial differential equations (PDEs).
Our method employs a Siamese network architecture that is motivated by the
mathematical properties of a metric. We leverage a controllable data generation
setup with PDE solvers to create increasingly different outputs from a
reference simulation in a controlled environment. 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 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 an adjustable training data difficulty and demonstrate the
robustness of LSiM via an evaluation on three real-world data sets.
