Mathematical model

A mathematical model is an abstract description of a concrete system. It uses mathematical concepts and language to represent how things work. By using math, researchers can study complex systems without needing the actual physical objects present. This process of creating these descriptions is called mathematical modeling. It is a vital tool used across many different fields. These include the natural sciences, social sciences, and engineering. It is also used in operations research to solve problems in business or military operations.

To build a model, researchers must identify several different types of variables. In business and engineering, models often aim to maximize a specific output. The system requires certain inputs to function. The relationship between these inputs and outputs depends on various factors. Decision variables, also called independent variables, are often controlled by the user. Exogenous variables act as parameters or constants. State variables are dependent on the decision, input, random, and exogenous variables. Finally, output variables depend on the current state of the system. Users often define objective functions to measure performance. These functions represent the specific goals or constraints of the system.

Mathematical models are classified into several distinct types based on their behavior. Linear models are those where all operators exhibit linearity. This structure allows a problem to be broken down into simpler, independent parts. Such problems can be analyzed at different scales while remaining valid. In contrast, nonlinear models involve nonlinear equations. Nonlinearity is often linked to complex phenomena like chaos and irreversibility. These systems are generally much more difficult to study than linear ones. One common approach is linearization, though this can fail when studying irreversible processes.

Models can also be categorized by how they handle time and change. A static model, or steady-state model, calculates a system in equilibrium. These models are time-invariant, meaning they do not change over time. A dynamic model accounts for changes that depend on time. These are typically represented using differential equations or difference equations. Models can also be described as discrete or continuous. Discrete models treat objects as separate entities, such as individual particles. Continuous models represent objects as a continuous flow, such as the velocity of fluid in a pipe.

Another way to classify models is by how much information is known beforehand. This is often described using the concepts of black-box and white-box models. A white-box model, also called a glass box, uses all available necessary information. These are often easier to use because they follow known rules. A black-box model is used when no a priori information is available. In these cases, researchers only observe the relationship between inputs and outputs. Many real-world systems fall somewhere between these two extremes. For example, a model of medicine in the blood might know the type of decay but not the exact rate. This requires estimating unknown parameters to make the model accurate.

Models are further divided by their relationship to certainty and logic. A deterministic model is one where every state is uniquely determined by its parameters. For a given set of initial conditions, a deterministic model will always perform the same way. Conversely, a stochastic model, or statistical model, includes randomness. In these models, variables are described by probability distributions rather than unique values. Models can also be deductive, based on a theory, or inductive, based on empirical findings. If a model rests on neither theory nor observation, it is called a floating model. Some strategic models in game theory are unique because they model agents with incompatible incentives. These models assume players are rational decision makers who maximize an objective function.

The relationship between theory and reality is a driving force in science. The quality of a scientific field often depends on how well theoretical models agree with repeatable experiments. When there is a lack of agreement between a model and experimental measurements, it is significant. This mismatch often leads to important advances as scientists develop better theories. In the physical sciences, a traditional model includes elements like governing equations, constitutive equations, and boundary conditions. By comparing math to the real world, we uncover the hidden mechanics of our universe.