  Many ontologies, in particular in the biomedical domain, are based on the
Description Logic EL++. Several efforts have been made to interpret and exploit
EL++ ontologies by distributed representation learning. Specifically, concepts
within EL++ theories have been represented as n-balls within an n-dimensional
embedding space. However, the intersectional closure is not satisfied when
using n-balls to represent concepts because the intersection of two n-balls is
not an n-ball. This leads to challenges when measuring the distance between
concepts and inferring equivalence between concepts. To this end, we developed
EL Box Embedding (ELBE) to learn Description Logic EL++ embeddings using
axis-parallel boxes. We generate specially designed box-based geometric
constraints from EL++ axioms for model training. Since the intersection of
boxes remains as a box, the intersectional closure is satisfied. We report
extensive experimental results on three datasets and present a case study to
demonstrate the effectiveness of the proposed method.
