Mathematical Models as Research Data via Flexiformal Theory Graphs

  • Michael KohlhaseEmail author
  • Thomas KopruckiEmail author
  • Dennis MüllerEmail author
  • Karsten TabelowEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)


Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open.

In this paper we propose a solution – to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexiformal framework, which has enough structure to support the various uses of models in scientific and technology workflows.

Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroeck’s drift-diffusion model for one-dimensional devices. This formalization – and future extensions – allows us to support the modeler by e.g., flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating mathematical models as “research data” and opens the way towards more MKM services for models.



We gratefully acknowledge EU funding for the OpenDreamKit project in the Horizon 2020 framework under grant 676541. Our discussions have particularly profited from contributions by Florian Rabe (Mmt advice) and Wolfram Sperber (general math background). Finally, Marcel Rupprecht has developed the MPD viewer mentioned in Sect. 4.


  1. [BGH05]
    Bandelow, U., Gajewski, H., Hünlich, R.: Fabry-perot lasers: thermodynamics-based modeling. In: Piprek, J. (ed.) Optoelectronic Devices. Springer, New York (2005)Google Scholar
  2. [Bra09]
    Brase, J.: DataCite - A global registration agency for research data. In: Fourth International Conference on Cooperation and Promotion of Information Resources in Science and Technology, COINFO 2009, pp. 257–261. IEEE (2009)Google Scholar
  3. [Far+16]
    Farrell, P., et al.: Numerical methods for drift-diffusion models. In: Piprek, J. (ed.) Handbook of Optoelectronic Device Modeling and Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Models, vol. 2. Taylor & Francis, Berlin (2016). (WIAS Preprint No. 2263. To appear, 2017)Google Scholar
  4. [Fey49]
    Feynman, R.P.: Space-time approach to quantum electrodynamics. Phys. Rev. 76(6), 769–789 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Fis+12]
    Fischer, A., et al.: Self-heating effects in organic semiconductor crossbar structures with small active area. Org. Electron. 13(11), 2461–2468 (2012)CrossRefGoogle Scholar
  6. [For15]
    Deutsche Forschungsgemeinschaft. DFG Guidelines on the Handling of Research Data. Adopted by the Senate of the DFG at September 30 (2015)Google Scholar
  7. [GS14]
    Greuel, G.-M., Sperber, W.: swMATH – an information service for mathematical software. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 691–701. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44199-2_103 Google Scholar
  8. [Gum64]
    Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electron Devices 11(10), 455–465 (1964)CrossRefGoogle Scholar
  9. [HKR12]
    Horozal, F., Kohlhase, M., Rabe, F.: Extending MKM formats at the statement leveld. In: Jeuring, J., Campbell, J.A., Carette, J., Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) CICM 2012. LNCS, vol. 7362, pp. 65–80. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31374-5_5 CrossRefGoogle Scholar
  10. [Huc+03]
    Hucka, M., et al.: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19(4), 524 (2003)CrossRefGoogle Scholar
  11. [Ian17]
    Iancu, M.: Towards flexiformal mathematics. Ph.D. thesis. Jacobs University, Bremen (2017)Google Scholar
  12. [Kas+16]
    Kaschura, F., et al.: Operation mechanism of high performance organic permeable base transistors with an insulated and perforated base electrode. J. Appl. Phys. 120(9), 094501 (2016)CrossRefGoogle Scholar
  13. [KI12]
    Kohlhase, M., Iancu, M.: Searching the Space of Mathematical Knowledge. In: Sojka, P., Kohlhase, M. (eds.) DML and MIR 2012. Masaryk University, Brno (2012).
  14. [Koh+]
    Kohlhase, M., et al.: A Case study for active documents and formalization in math models: the van Roosbroeck Model. Accessed 5 Feb 2017
  15. [Koh06]
    Kohlhase, M.: OMDoc – An Open Markup Format for Mathematical Documents [version 1.2]. LNCS (LNAI), vol. 4180. Springer, Heidelberg (2006). CrossRefGoogle Scholar
  16. [Koh13]
    Kohlhase, M.: The flexiformalist manifesto. In: Voxronkov, A., et al. (eds.) 14th International Workshop on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2012), pp. 30–36. IEEE Press, Timisoara (2013).
  17. [Kra15]
    Kraft, A.: RADAR-A repository for long tail data. In: Proceedings of the IATUL Conferences. Paper 1 (2015)Google Scholar
  18. [KT16]
    Koprucki, T., Tabelow, K.: Mathematical models: a research data category? In: Greuel, G.-M., Koch, T., Paule, P., Sommese, A. (eds.) ICMS 2016. LNCS, vol. 9725, pp. 423–428. Springer, Cham (2016). doi: 10.1007/978-3-319-42432-3_53 CrossRefGoogle Scholar
  19. [Mie11]
    Mielke, A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Mit]
    MitM: The Math-in-the-Middle Ontology. Accessed 5 Feb 2017
  21. [Mod14]
    Modelica Association. Modelica-A Unified Object-Oriented Language for Physical Systems Modeling-Language Specification Version 3.3 Revision 1, 2014 (2014).
  22. [MPDHub]
    MPDHub wiki. Accessed 22 Mar 2017
  23. [Pfe01]
    Pfenning, F.: Logical Frameworks. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, Vol. I and II. Elsevier Science and MIT Press, North Holland (2001)Google Scholar
  24. [Rab]
  25. [Rab14]
    Rabe, F.: How to identify, translate, and combine logics? J. Log. Comput. (2014)Google Scholar
  26. [RK13]
    Rabe, F., Kohlhase, M.: A scalable module system. Inf. Comput. 230, 1–54 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  27. [RNH14]
    Razum, M., Neumann, J., Hahn, M.: RADAR-Ein Forschungsdaten- repositorium als Dienstleistung für die Wissenschaft. Z. Bibl. Bibliographie 61(1), 18–27 (2014)CrossRefGoogle Scholar
  28. [SBML]
    The Systems Biology Markup Language. Accessed 17 Mar 2017
  29. [Sch96]
    Schröter, J.: Zur Meta-Theorie der Physik. Walter de Gruyter GmbH & Co KG, Berlin (1996)CrossRefGoogle Scholar
  30. [Sel84]
    Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Wien, New York (1984)CrossRefGoogle Scholar
  31. [UML]
    Unified Modeling Language. Accessed 13 Sep 2016
  32. [VR50]
    Van Roosbroeck, W.: Theory of the flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29(4), 560–607 (1950)CrossRefGoogle Scholar
  33. [Wik]
    Wikipedia: List of physical quantities – Wikipedia, The Free Encyclopedia. Accessed 22 Mar 2017

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Informatik, FAU Erlangen-NürnbergErlangenGermany
  2. 2.Weierstrass Institute (WIAS)BerlinGermany

Personalised recommendations