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Principles of Mathematical Modeling

We begin this introduction to mathematical modeling and simulation with an explanation of basic concepts and ideas, which includes definitions of terms such as system, model, simulation, and mathematical model, reflections on the objectives of mathematical modeling and simulation, characteristics of "good" mathematical models, and a classification of mathematical models. You may skip this chapter at first reading if you are just interested in a hands-on application of specific methods explained in the later chapters of the book, such as regression or neural network methods (Chapter 2), differential equations (DEs) (in Chapters 3 and 4), or virtual plants (Chapter 5). Any professional in this field, however, should of course know about the principles of mathematical modeling and simulation. It was emphasized in the preface that everybody uses mathematical models?-?"even those of us who are not aware of doing so". You will agree that it is a good idea to have an idea of what one is doing.

Our starting point is the complexity of the problems treated in science and engineering. As will be explained in Section 1.1, the difficulty of problems treated in science and engineering typically originates from the complexity of the systems under consideration, and models provide an adequate tool to break up this complexity and make a problem tractable. After giving general definitions of the terms system, model, and simulation in Section 1.2, we move on toward mathematical models in Section 1.3, where it is explained that mathematics is the natural modeling language in science and engineering. Mathematical models themselves are defined in Section 1.4, followed by a number of example applications and definitions in Sections 1.5 and 1.6. This includes the important distinction between phenomenological and mechanistic models, which has been used as the main organization principle of this book (see Section 1.6.1 and Chapters 2-5). The chapter ends with a classification of mathematical models and Golomb's famous "Don'ts of mathematical modeling" in Sections 1.7 and 1.8.

1.1 A Complex World Needs Models

Generally speaking, engineers and scientists try to understand, develop, or optimize " systems". Here, "system" refers to the object of interest, which can be a part of nature (such as a plant cell, an atom, a galaxy, etc.) or an artificial technological system (see Definition 1.2.3). Principally, everybody deals with systems in their everyday life in a way similar to the approach of engineers or scientists. For example, consider the problem of a table that is unstable due to an uneven floor. This is a technical system, and everybody knows what must be done to solve the problem: we just have to put suitable pieces of cardboard under the table legs. Each of us solves an abundant number of problems relating to simple technological systems of this kind during our lifetime. Beyond this, there are a great number of really difficult technical problems that can only be solved by engineers. Characteristic of these more demanding problems is a high complexity of the technical system. We would simply need no engineers if we did not have to deal with complex technical systems such as computer processors, engines, and so on. Similarly, we would not need scientists if processes such as the photosynthesis of plants could be understood as simply as an unstable table. The reason why we have scientists and engineers, virtually their right to exist, is the complexity of nature and the complexity of technological systems.

Note 1.1.1 (The complexity challenge) It is the genuine task of scientists and engineers to deal with complex systems, and to be effective in their work, they most notably need specific methods to deal with complexity.

The general strategy used by engineers or scientists to break up the complexity of their systems is the same strategy that we all use in our everyday life when we are dealing with complex systems: simplification. The idea is just this: if something is complex, make it simpler. Consider an everyday life problem related to a complex system: A car that refuses to start. In this situation, everyone knows that a look at the battery and fuel levels will solve the problem in most cases. Everyone will do this automatically, but to understand the problem-solving strategy behind this, let us think of an alternative scenario. Assume someone is in this situation for the first time. Assume that "someone" was told how to drive a car, that they have used the car for some time, and now they are for the first time in a situation in which the car does not start. Of course, we also assume that there is no help for miles around! Then, looking under the hood for the first time, our "someone" will realize that the car, which seems simple as long as it works well, is quite a complex system. They will spend a lot of time until they eventually solve the problem, even if we admit that our "someone" is an engineer. The reason why each of us will solve this problem much faster than this "someone" is of course the simple fact that this situation is not new to us. We have experienced this situation before, and from our previous experience we know what is to be done. Conceptually, one can say that we have a simplified picture of the car in our mind similar to Figure 1.1. In the moment when we realize that our car does not start, we do not think of the car as the complex system that it really is, that is, we do not think of this conglomerate of valves, pistons, and all the kind of stuff that can be found under the hood; rather, we have this simplified picture of the car in our mind. We know that this simplified picture is appropriate in this given situation, and it guides us to look at the battery and fuel levels and then solve the problem within a short time.

Figure 1.1 Car as a real system and as a model.

This is exactly the strategy used by engineers or scientists when they deal with complex systems. When an engineer, for example, wants to reduce the fuel consumption of an engine, they will not consider that engine in its entire complexity. Rather, they will use simplified descriptions of that engine, focusing on the machine parts that affect fuel consumption. Similarly, a scientist who wants to understand the process of photosynthesis will use simplified descriptions of a plant focusing on very specific processes within a single plant cell. Anyone who wants to understand complex systems or solve problems related to complex systems needs to apply appropriate simplified descriptions of the system under consideration. This means that anyone who is concerned with complex systems needs models, since simplified descriptions of a system are models of that system by definition.

Note 1.1.2 (Role of models) To break up the complexity of a system under consideration, engineers and scientists use simplified descriptions of that system (i.e. models).

1.2 Systems, Models, Simulations

In 1965, Minsky gave the following general definition of a model [2, 3]:

Definition 1.2.1 (Model) To an observer B, an object is a model of an object A to the extent that B can use to answer questions that interest him about A.

Note 1.2.1 (Formal definitions) Note that Definition 1.2.1 is a formal definition in the sense that it operates with terms such as object or observer that are not defined in a strict axiomatic sense similar to the terms used in the definitions of standard mathematical theory. The same remark applies to several other definitions in this book, including the definition of the term mathematical model in Section 1.4. Definitions of this kind are justified for practical reasons, since they allow us to talk about the formally defined terms in a concise way. An example is Definition 2.6.2 in Section 2.6.5, a concise formal definition of the term overfitting, which uses several of the previous formal definitions.

The application of Definition 1.2.1 to the car example is obvious?-?we just have to identify B with the car driver, A with the car itself, and with the simplified tank/battery description of the car in Figure 1.1.

1.2.1 Teleological Nature of Modeling and Simulation

An important aspect of the above definition is the fact that it includes the purpose of a model, namely, that the model helps us to answer questions and solve problems. This is important because particularly beginners in the field of modeling tend to believe that a good model is one that mimics the part of reality that it pertains to as closely as possible. But as was explained in the previous section, modeling and simulation aim at simplification, rather than at a useless production of complex copies of a complex reality, and hence, the contrary is true.

Note 1.2.2 (The best model) The best model is the simplest model that still serves its purpose, that is, which is still complex enough to help us understand a system and to solve problems. Seen in terms of a simple model, the complexity of a complex system...