Mathematical Modelling by Help of Category Theory: Models and Relations between Them

Abstract

The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios.

Mathematical Modeling Research Output Impacting New Technological Development: An Axiomatization to Build Novelty

“…The current study presents categorical mathematical modeling output in the form of mathematical formulae [15]. However, it is unclear to determine and understand the modeling output, modeling purposes and its impact on new technological development.…”

“…Due to the abstractness of mathematical modeling output, the modeling objective can be unclear on how it impacts the development of new technologies [18] (i.e., the use of specific mathematical modeling to produce new technology). Research carried out by [15] proposed an abstract category using a theorybased approach to mathematical modeling research output; however, it does not provide qualitative elements of the categories nor the impacts on new technology development.…”

“…Hence, obtaining scientific support in designing the optimal path of mathematical modeling that impacts new technology development programs is of paramount importance under the certain limit of modeling categorization. An interesting previous work by [15] demonstrates a formulation of mathematical model categorization using a theory-based approach. Following this method, mathematical models are formalized in the mathematical language of categories, and relations between the models are formally defined.…”

Digital twin: Data exploration, architecture, implementation and future

“…However, each modeling language has its meaning, which makes it hard to exchange data and formats that are compatible with other languages. New mathematical approaches based on Category theory [50] could provide a complete foundation for modeling, interoperability, and integration. Based on this, the data can then be saved.…”

Preface to the Special Issue on “Modelling and Simulation in Engineering”

“…The proposed high-dimensional Kriging modelling method is characterised by a faster modelling efficiency and large possibilities for new applications. D. Legatiuk proposes in [14] a new development of the category theory-based modelling methodology introduced in a previous paper that he co-authored. This methodology is based on representation of mathematical models by the help of categorical constructions.…”