  OWL ontologies, whose formal semantics are rooted in Description Logic (DL),
have been widely used for knowledge representation. Similar to Knowledge Graphs
(KGs), ontologies are often incomplete, and maintaining and constructing them
has proved challenging. While classical deductive reasoning algorithms use the
precise formal semantics of an ontology to predict missing facts, recent years
have witnessed growing interest in inductive reasoning techniques that can
derive probable facts from an ontology. Similar to KGs, a promising approach is
to learn ontology embeddings in a latent vector space, while additionally
ensuring they adhere to the semantics of the underlying DL. While a variety of
approaches have been proposed, current ontology embedding methods suffer from
several shortcomings, especially that they all fail to faithfully model
one-to-many, many-to-one, and many-to-many relations and role inclusion axioms.
To address this problem and improve ontology completion performance, we propose
a novel ontology embedding method named Box$^2$EL for the DL EL++, which
represents both concepts and roles as boxes (i.e., axis-aligned
hyperrectangles), and models inter-concept relationships using a bumping
mechanism. We theoretically prove the soundness of Box$^2$EL and conduct an
extensive experimental evaluation, achieving state-of-the-art results across a
variety of datasets on the tasks of subsumption prediction, role assertion
prediction, and approximating deductive reasoning.
