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Towards a Knowledge Graph for Models and Algorithms in Applied Mathematics

Björn Schembera, Frank Wübbeling, Hendrik Kleikamp, Burkhard Schmidt, Aurela Shehu, Marco Reidelbach, Christine Biedinger, Jochen Fiedler, Thomas Koprucki, Dorothea Iglezakis, Dominik Göddeke

TL;DR

This work tackles the challenge of representing mathematical models and numerical algorithms as interoperable, FAIR research data by merging two prior ontologies (MathModDB and MathAlgoDB) into a living knowledge graph linked through Computational Tasks. It introduces a quantities-centric semantic framework with new metadata enrichment, including QuantityKind, and establishes mechanisms for model–algorithm coupling via explicit relationships (requires, recommends, precludes), enabling automatic algorithm selection via SPARQL. Demonstrations with use cases such as gravity-driven free fall and a Romanization spread model show how the KG supports modeling workflows and parameter inference, while the system already contains over 2000 elements and more than 250 assets; MaRDMO integration and a web/SPARQL interface underscore production readiness. The approach promises broad applicability across applied mathematics and related domains, with ongoing work to automate data ingestion and extend coverage to additional mathematical domains, while acknowledging current discretization handling limitations.

Abstract

Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.

Towards a Knowledge Graph for Models and Algorithms in Applied Mathematics

TL;DR

This work tackles the challenge of representing mathematical models and numerical algorithms as interoperable, FAIR research data by merging two prior ontologies (MathModDB and MathAlgoDB) into a living knowledge graph linked through Computational Tasks. It introduces a quantities-centric semantic framework with new metadata enrichment, including QuantityKind, and establishes mechanisms for model–algorithm coupling via explicit relationships (requires, recommends, precludes), enabling automatic algorithm selection via SPARQL. Demonstrations with use cases such as gravity-driven free fall and a Romanization spread model show how the KG supports modeling workflows and parameter inference, while the system already contains over 2000 elements and more than 250 assets; MaRDMO integration and a web/SPARQL interface underscore production readiness. The approach promises broad applicability across applied mathematics and related domains, with ongoing work to automate data ingestion and extend coverage to additional mathematical domains, while acknowledging current discretization handling limitations.

Abstract

Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.
Paper Structure (13 sections, 1 equation, 3 figures, 1 table)

This paper contains 13 sections, 1 equation, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Merging MathModDB and MathAlgoDB
  4. Previous Ontology Structures and their Shortcomings
  5. Computational Tasks as the missing Link
  6. Quantities are central to semantic Knowledge Representation
  7. Metadata Enrichment for mathematical Models
  8. Metadata Enrichtment for mathematical Algorithms
  9. Living Knowledge Graph of Models & Algorithms
  10. Use Case: From falling Apples to moving Planets
  11. Use Case: Like a Virus - Romanization of Northern Tunesia
  12. Data Flows in the Knowledge Graph
  13. Conclusion & Outlook

Figures (3)

  • Figure 1: Essential classes of the related work ontologies for mathematical models MathModDB (a) and numerical algorithms MathAlgoDB (b) from schembera2024ontologies.
  • Figure 2: Schematic display of the joint ontology for mathematical models and algorithms. Classes and important relations are depicted.
  • Figure 3: Schematic display of the sample use case \ref{['sec:UseCaseApple']} free fall model in pomology in the joint ontology. For reasons of clarity, we will limit ourselves here to the essential entries and not show all connections, classes and quantities.