FAIR Representation of Mathematical Research Data: MathModDB and MathAlgoDB as Knowledge Graphs for Mathematical Models and Numerical Algorithms

Important components of mathematical research data are mathematical models, for the application of mathematical methods to real-world problems, and mathematical algorithms, for data processing. We use semantic technologies, i.e., ontologies and knowledge graphs (KG), to establish metadata for these complex research data, thus making them FAIR. Developed within the scope of the Mathematical Research Data Initiative (MaRDI) project, a first draft of ontologies for models (MathModDB) and algorithms (MathAlgoDB) was presented at the CoRDI 2023. This work was refined to produce a more advanced version, featuring stable classes, consistent naming, metadata enrichment and numerous models and algorithms integrated as data, thereby paving the way towards an actionable KG. MathModDB was initially developed in RDF format using the Protégé ontology editor, whereas MathAlgoDB was created within an Apache Jena/Django framework. Stable versions of the ontologies and their data were released on GitHub and Zenodo. The published versions are provided with a Widoco documentation. MaRDMO – a plugin for the Research Data Management Organiser – allows researchers to contribute structured metadata to the MathAlgoDB and MathModDB KGs and to retrieve existing data for reuse. The latest conceptual change is the introduction of interface classes semantically linking the two graphs, with a Computational Task in MathModDB corresponding to an Algorithmic Task in MathAlgoDB. This represents how a concrete computational task can be solved algorithmically. A single mathematical model may lead to different tasks depending on the choice of known or unknown quantities. In the example of a model representing emission tomography without scatter, the attenuation distribution can be either known or unknown, resulting in two different computational tasks. These are equivalent to two different algorithmic tasks, each of which is handled by different solution algorithms. MathModDB was exported to the MaRDI Portal in February 2025, integrating the model database into the comprehensive MaRDI KG. The MaRDI Portal serves as an one-stop-shop for FAIR mathematical data and is based on a customized Wikibase instance. The export was accomplished with the help of a dictionary that mapped the RDF ontology objects to portal items, prioritizing the reuse of existing Wikidata items and creating new ones if necessary. One significant advantage of the MaRDI Portal is the ability to easily use qualifiers, which enable the construction of hierarchies of mathematical models potentially specifying the assumptions under which the models are interconnected. Linking mathematical formulas with quantities via qualifiers improves human and machine readability. The use of more generic object properties with qualifiers increases the depth of the semantic description and can simplify the semantic integration of other systems. This new KG makes models and algorithms FAIR and machine-actionable, enabling reuse by other NFDI consortia. To support this, we aim to integrate it into the NFDI Core Ontology and related process ontologies , with a focus on (semi-) automated model and metadata capture in future work.