  Ontologies contain rich knowledge within domain, which can be divided into
two categories, namely extensional knowledge and intensional knowledge.
Extensional knowledge provides information about the concrete instances that
belong to specific concepts in the ontology, while intensional knowledge
details inherent properties, characteristics, and semantic associations among
concepts. However, existing ontology embedding approaches fail to take both
extensional knowledge and intensional knowledge into fine consideration
simultaneously. In this paper, we propose a novel ontology embedding approach
named EIKE (Extensional and Intensional Knowledge Embedding) by representing
ontologies in two spaces, called extensional space and intensional space. EIKE
presents a unified framework for embedding instances, concepts and their
relations in an ontology, applying a geometry-based method to model extensional
knowledge and a pretrained language model to model intensional knowledge, which
can capture both structure information and textual information. Experimental
results show that EIKE significantly outperforms state-of-the-art methods in
three datasets for both triple classification and link prediction, indicating
that EIKE provides a more comprehensive and representative perspective of the
domain.
