The modeling process can be viewed as a transition from one class of models to another. Three broad classes of models are conventionally distinguished: cognitive, substantive, and formal. They constitute three interrelated levels of modeling that cannot be considered in isolation. The mutual influence between modeling levels is connected with the property of potentiality inherent in models. The creation of any model entails the emergence of new knowledge about the object under study, leading to the reassessment and refinement of concepts and views of the modeling object at different levels. This circumstance, in turn, leads to a revision of the corresponding substantive and cognitive models, ensuring a spiral development of all levels of modeling of the object under study.
When observing the original object, a mental image of the object — its ideal model — forms in the researcher's mind. In the scientific literature, this is commonly called a cognitive (mental) model. In forming such a model, the researcher, as a rule, seeks to answer specific questions; therefore, everything inessential is stripped from the object's infinitely complex makeup to obtain a more compact, concise description. The representation of a cognitive model in natural language is called a substantive model.
Cognitive models are inherently subjective, as they are constructed by researchers based on prior knowledge and experience. These models can only be understood through symbolic description. Cognitive and substantive models are not equivalent, since the former may contain elements that the researcher is unable or unwilling to articulate. At the same time, if a substantive model has been formulated by someone else or is the product of collective effort, its interpretation, the level of understanding, and the degree of trust may vary significantly among interpreters. Substantive models are often referred to as technical problem statements.
By functional purpose and objectives, substantive models are divided into descriptive, explanatory, and predictive:
- A descriptive model is any description of an object.
- An explanatory model answers the question of why something happens.
- A predictive model describes an object's future behavior. Notably, a predictive model does not necessarily include an explanatory component.
A conceptual model is a substantive model whose formulation employs the concepts and representations of the subject domains concerned with the study of the modeling object.
In a broader sense, a conceptual model is understood as a substantive model based on a specific concept or point of view. Three types of conceptual models are distinguished: logical-semantic, structural-functional, and causal.
- A logical-semantic model describes an object using the terms and definitions of relevant subject domains, incorporating all known logically consistent statements and facts. Analysis of these models relies on logical reasoning and the accumulated knowledge within the respective domains.
- A structural-functional model treats the object as an integrated system, decomposed into individual elements or subsystems. The components are connected by structural relations that define subordination, logical sequencing, and the temporal order of task completion. Diagrams, charts, and maps are commonly used to represent these models.
- A causal model is often used to explain and predict the behavior of an object. These models are oriented primarily toward: (1) identifying the key interrelationships between the constituent elements of the object under study; (2) determining how changes in certain factors affect the state of the model's components; (3) understanding how the model as a whole will function and whether it will adequately describe the dynamics of the parameters of interest to the researcher.
A formal model represents a conceptual model using one or more formal languages, such as mathematical theories, specialized modeling languages, or programming languages. In the humanities, modeling often concludes with the development of a conceptual model, whereas in the natural sciences, constructing a formal model is generally feasible.
Although the importance of substantive and formal models in research is widely acknowledged, the role of cognitive models is often underestimated due to their subjectivity and the implicit nature of thought processes. Nevertheless, cognitive models are particularly significant in contexts where operators or decision-makers rely on their own mental representations to control objects or processes. This type of model also plays a substantial role in the social sciences.
Mathematical modeling is a form of scientific, symbolic, and formal modeling in which objects are described using mathematical language and investigated through mathematical methods.
A mathematical model designed for scientific research enables the determination of parameter values of the modeled object or phenomenon based on specified input data. Thus, the core of such a model is the mapping of input parameter values to output parameter values. This perspective allows a mathematical model to be viewed as a mathematical operator, which may be broadly defined as a function, a mapping represented by algebraic, differential, integro-differential, or integral equations, an algorithm, a set of rules, or tables that facilitate the determination of output parameters from input values.
Defining a mathematical model through the concept of an operator is more constructive from the standpoint of building a classification of such models, since it encompasses the full diversity of mathematical models that exist today.
Advantages of mathematical modeling compared with physical experiment:
- economy (in particular, conservation of the real system's resources);
- the ability to model hypothetical objects — that is, objects not realized in nature;
- the ability to simulate operating conditions that are dangerous or difficult to reproduce in reality;
- the ability to change the time scale;
- ease of multi-faceted analysis;
- greater predictive power, owing to the ability to identify general patterns;
- universality of the hardware and software used (programming systems and general-purpose application software packages).
An information model is a model of an object presented in the form of information describing the parameters and variables essential for the given analysis, the relationships between them, and the inputs and outputs of the object, which allows one to simulate possible states of the object by feeding information about changes in input values into the model.