  Ontology embeddings map classes, relations, and individuals in ontologies
into $\mathbb{R}^n$, and within $\mathbb{R}^n$ similarity between entities can
be computed or new axioms inferred. For ontologies in the Description Logic
$\mathcal{EL}^{++}$, several embedding methods have been developed that
explicitly generate models of an ontology. However, these methods suffer from
some limitations; they do not distinguish between statements that are
unprovable and provably false, and therefore they may use entailed statements
as negatives. Furthermore, they do not utilize the deductive closure of an
ontology to identify statements that are inferred but not asserted. We
evaluated a set of embedding methods for $\mathcal{EL}^{++}$ ontologies based
on high-dimensional ball representation of concept descriptions, incorporating
several modifications that aim to make use of the ontology deductive closure.
In particular, we designed novel negative losses that account both for the
deductive closure and different types of negatives. We demonstrate that our
embedding methods improve over the baseline ontology embedding in the task of
knowledge base or ontology completion.
