Title: NeuroSED: Learning Subgraph Similarity via Graph Neural Networks. Abstract: Subgraph similarity search is a fundamental operator in graph analysis. In this framework, given a query graph and a graph database, the goal is to identify subgraphs of the database graphs that are structurally similar to the query. Subgraph edit distance (SED) is one of the most expressive measures of subgraph similarity. In this work, we study the problem of learning SED from a training set of graph pairs and their SED values. Towards that end, we design a novel siamese graph neural network called NeuroSED, which learns an embedding space with a rich structure reminiscent of SED. With the help of a specially crafted inductive bias, NeuroSED not only enables high accuracy but also ensures that the predicted SED, like true SED, satisfies triangle inequality. The design is generic enough to also model graph edit distance (GED), while ensuring that the predicted GED space is metric, like the true GED space. Extensive experiments on real graph datasets, for both SED and GED, establish that NeuroSED achieves $\approx 2$ times lower RMSE than the state of the art and is $\approx 18$ times faster than the fastest baseline. Further, owing to its pair-independent embeddings and theoretical properties, NeuroSED allows orders-of-magnitude faster graph/subgraph retrieval.