Mathematical Model

Description of a system using mathematical concepts and language

For other uses, see Mathematical model (disambiguation).

A mathematical model, in its most fundamental sense, serves as an abstract representation—a conceptual blueprint, if you will—of a tangible, real-world system. This description is meticulously constructed using the precise language and rigorous concepts inherent to mathematics. The entire endeavor of conceptualizing, formulating, and refining such a model is aptly termed mathematical modeling. These models are not mere academic curiosities; they permeate an astonishing breadth of disciplines, finding indispensable application across applied mathematics, the vast expanse of the natural sciences, the intricate webs of the social sciences 1 2, and the pragmatic realm of engineering. Notably, the specialized field of operations research dedicates itself to the systematic study and deployment of mathematical modeling, alongside its associated analytical tools, to disentangle and resolve complex challenges encountered in both commercial enterprises and military logistical operations.

Fundamentally, a well-crafted model offers a lens through which to characterize a system. By dissecting and scrutinizing the individual contributions and interactions of its various components, researchers can glean profound insights. This analytical process is not merely descriptive; it empowers practitioners to formulate educated predictions concerning the system's future behavior or to devise optimal solutions for specific, pressing problems. One might even say it allows us to anticipate the universe's next disappointing plot twist.

Elements of a mathematical model

The forms that mathematical models can assume are as varied and complex as the systems they endeavor to describe. They might manifest as intricate dynamical systems, sophisticated statistical models, precise differential equations, or the strategic frameworks of game theoretic models. It's crucial to understand that these categories are rarely mutually exclusive; a single, comprehensive model often weaves together a diverse tapestry of these abstract structures.

In the rigorous pursuit of scientific understanding, the true mettle of any theoretical construct is tested by its concordance with empirical reality. Consequently, the perceived quality and advancement of a scientific field are often directly proportional to how closely its theoretically derived mathematical models align with the outcomes of repeatable experiments. When a glaring discrepancy emerges between the elegant predictions of a theoretical mathematical model and the stubborn, undeniable results of experimental measurements, it's rarely a sign of failure. Instead, such disagreements frequently act as powerful catalysts, driving significant advancements as scientists are compelled to develop better, more nuanced theories that can account for the observed phenomena. It's almost as if reality enjoys proving us wrong, forcing us to refine our understanding.

In the hallowed halls of the physical sciences, a traditional, robust mathematical model typically encompasses most, if not all, of the following foundational elements:

Governing equations: The fundamental laws or principles that dictate the system's behavior.
Supplementary sub-models: Smaller, often specialized models that describe specific components or processes within the larger system, acting as intricate nested dolls of abstraction.
Defining equations: Equations that establish the fundamental identities or relationships between variables.
Constitutive equations: Equations that describe the response of a material or system to external stimuli, essentially defining its "personality."
Assumptions and constraints: The often-unspoken (or begrudgingly stated) boundaries and simplifications within which the model is expected to operate, acknowledging that reality is far too messy to model perfectly.
Initial and boundary conditions: The starting points and edges of the system, crucial for defining its specific behavior over time or space.
Classical constraints and kinematic equations: Principles governing the motion of objects without considering the forces causing that motion, providing the skeletal framework of movement.

Classifications

Mathematical models are not a monolithic entity; they come in various types, each suited to capturing different facets of the systems they represent. Understanding these classifications is less about rigid categorization and more about appreciating the spectrum of tools available to the perpetually unsatisfied modeler.

Linear vs. nonlinear

The distinction between linear and nonlinear models is a fundamental one, often dictating the ease (or abject difficulty) of analysis. If every single operator within a mathematical model demonstrates the property of linearity, then the resulting mathematical model is, by definition, linear. All other models, by default, are relegated to the considerably more complex and often chaotic realm of nonlinear models.

It's worth noting that the precise definition of linearity and nonlinearity can, rather inconveniently, depend on the specific context. Consequently, what might be deemed a "linear model" can, paradoxically, contain expressions that are inherently nonlinear. For instance, in the domain of a statistical linear model, the core assumption is that a relationship is linear in its parameters, even if the relationship itself is decidedly nonlinear with respect to the predictor variables. Similarly, a differential equation earns the label "linear" if it can be expressed using only linear differential operators, yet it might still harbor nonlinear expressions within its structure. When it comes to mathematical programming models, if both the objective functions and the constraints can be represented entirely by linear equations, then the model is comfortably classified as linear. However, the moment even a single objective function or constraint is characterized by a nonlinear equation, the model immediately transcends into the more challenging territory of a nonlinear model.

The allure of a linear structure lies in its inherent decomposability. A linear problem can often be broken down into simpler, more manageable parts, which can then be analyzed independently or at different scales. Crucially, the results derived from these simpler parts or scaled analyses are expected to remain valid when recomposed or rescaled, offering a gratifying sense of order.

Nonlinearity, on the other hand, even within ostensibly simple systems, is frequently the harbinger of complex and often counterintuitive phenomena. Think chaos, the butterfly effect, and the stubborn reality of irreversibility. With rare exceptions, nonlinear systems and their corresponding models tend to be significantly more arduous to study and understand than their linear counterparts. A common, albeit often precarious, strategy for tackling nonlinear problems is linearization—approximating the nonlinear system with a linear one over a small operating range. This approach, however, can be deeply problematic if the very aspects one seeks to investigate, such as irreversibility or emergent chaotic behavior, are inextricably linked to the underlying nonlinearity. It's like trying to understand a storm by only studying a single raindrop.

Static vs. dynamic

The temporal dimension introduces another crucial classification. A dynamic model is designed to explicitly account for changes in the system's state that unfold over time, capturing its evolution and transient behaviors. Conversely, a static model (often referred to as a steady-state model) focuses solely on calculating the system's behavior when it has reached a state of equilibrium, where all time-dependent changes have ceased, rendering it time-invariant. Dynamic models are most commonly articulated through the powerful mathematical frameworks of differential equations or difference equations, which inherently describe rates of change or discrete steps through time.

Explicit vs. implicit

The nature of how a model's outputs are derived from its inputs defines whether it is explicit or implicit. If all the necessary input parameters for the overall model are precisely known, and the corresponding output parameters can be directly computed through a finite, sequential series of calculations, then the model is deemed explicit. This is the straightforward path, where cause-and-effect are clearly delineated and computed.

However, the world is rarely so obliging. Sometimes, it is the output parameters that are known, and the challenge lies in determining the corresponding inputs that must have given rise to them. In such scenarios, the solution cannot be directly calculated but must be iteratively sought through numerical procedures, such as the venerable Newton's method or the more sophisticated Broyden's method. When this iterative approach is necessary, the model is classified as implicit.

Consider, for example, the intricate workings of a jet engine. Its fundamental physical properties—like the precise dimensions of its turbine and nozzle throat areas—can be explicitly calculated if you are given a specific design thermodynamic cycle (encompassing air and fuel flow rates, pressures, and temperatures) at a particular flight condition and power setting. The relationship here is direct. Yet, the engine's operating cycles at other flight conditions and power settings cannot be explicitly derived solely from those constant physical properties. Instead, one must work backward, implicitly solving for the conditions that would yield a desired output, a task requiring considerably more computational finesse.

Discrete vs. continuous

The way a model conceptualizes the entities within a system forms the basis of another fundamental distinction. A discrete model treats its constituent objects as distinct, separable units. Think of individual particles in a molecular model, or the distinct, quantifiable states within a statistical model. These are countable, often distinct entities.

In stark contrast, a continuous model represents the objects and properties of a system as smoothly varying, unbroken quantities. Examples abound: the seamless velocity field of fluid flowing through a pipe, the gradual gradients of temperatures and stresses within a solid material, or the pervasive electric field that continuously permeates an entire region due to a point charge. Here, change is fluid, without abrupt jumps or gaps, reflecting a different philosophical approach to describing reality.

Deterministic vs. probabilistic (stochastic)

This classification hinges on the role of chance within a model's operation. A deterministic model is one where every single set of variable states is uniquely and unequivocally determined by the model's parameters and by the previous states of those variables. The implication is profound: for any given set of initial conditions, a deterministic model will always produce the exact same outcome, performing with predictable, clockwork precision. There is no room for randomness, no roll of the dice.

Conversely, a stochastic model—more commonly, and perhaps more accurately, termed a "statistical model"—embraces the inherent messiness of randomness. In such a model, variable states are not pinned down by unique, fixed values; instead, they are described by probability distributions. This means that even with identical initial conditions, repeated runs of a stochastic model will likely yield different, though statistically related, outcomes. It acknowledges that the universe, or at least our understanding of it, often prefers a range of possibilities over a single, predetermined fate.

Deductive, inductive, or floating

The intellectual genesis of a model is another important differentiator. A deductive model is a logical construct, meticulously built upon an established theory or a set of foundational principles. It proceeds from the general to the specific, drawing conclusions that are necessarily true if the premises are true. An inductive model, on the other hand, emerges from the careful observation of empirical findings, generalizing from specific instances to broader patterns. It's built from the ground up, inferring principles from data.

Then there are the 'floating' models—a rather unsettling classification for those who prefer their intellectual structures to be firmly anchored. A model is described as 'floating' if it appears to rest on neither a coherent, pre-existing theory nor a solid foundation of empirical observation. This lack of grounding has drawn pointed criticism, particularly regarding the application of mathematics in certain social sciences outside of economics, where models are sometimes perceived as theoretical castles built on sand 3. The application of catastrophe theory in various scientific contexts, for instance, has been famously characterized as a 'floating model' due to its perceived detachment from underlying mechanisms 4. One might say they are models in search of a purpose, or perhaps just models enjoying the view from their ivory tower.

Strategic vs. non-strategic

Models employed within the specialized domain of game theory possess a distinctive characteristic: they are specifically designed to model agents (or "players") whose incentives are inherently incompatible or even directly opposed. Think of competing species vying for limited resources, or bidders locked in a fierce auction. These are not systems where everyone is pulling in the same direction.

Strategic models operate on the fundamental assumption that these players are autonomous decision-makers, each acting rationally to maximize their own individual objective function. The core challenge in utilizing strategic models lies in the intricate task of defining and subsequently computing solution concepts, such as the celebrated Nash equilibrium, which represents a state where no player can improve their outcome by unilaterally changing their strategy. An intriguing and powerful property of strategic models is their ability to neatly separate the reasoning about the fundamental rules of the "game" from the complex reasoning about the actual behavior of the players within those rules 5. It's a useful distinction, assuming, of course, that "rationality" is a constant in human affairs, which is, shall we say, a generous assumption.

Construction

In the pragmatic arenas of business and engineering, the primary motivation for developing mathematical models is often the pursuit of optimization—specifically, to maximize a certain desired output or minimize an undesirable one. Any system under consideration in such contexts will, by its very nature, demand certain inputs. The intricate relationship that transforms these inputs into outputs is, however, not solely dependent on the inputs themselves. It is also influenced by a host of other critical variables: decision variables, which are under the control of the modeler; state variables, which describe the internal condition of the system; exogenous variables, which originate outside the system and influence it; and random variables, which introduce an element of unpredictable chance.

Decision variables are sometimes, rather tellingly, referred to as independent variables, highlighting their role as levers the modeler can manipulate. Exogenous variables, conversely, are occasionally labeled as parameters or constants, underscoring their fixed or external nature. It's crucial to recognize that these variables are rarely truly independent of one another. The state variables, for instance, are invariably dependent on the interplay of decision, input, random, and exogenous variables. Furthermore, the ultimate output variables are, by definition, contingent upon the current state of the system, as encapsulated by its state variables.

The desired objectives and inherent constraints of the system, along with the specific goals of its users, can all be elegantly represented as functions of either the output variables or the state variables. The precise form of these objective functions will, naturally, hinge entirely on the perspective and priorities of the model's user. Depending on the specific context, an objective function might also be known as an "index of performance," given that it serves as a quantifiable measure of interest to the user. While there is no theoretical limit to the number of objective functions and constraints a model can incorporate, it's a cold, hard truth that as this number escalates, the process of using or optimizing the model becomes exponentially more involved and computationally intensive. For example, economists frequently employ the tools of linear algebra when constructing and analyzing input–output models to understand inter-industry relationships. For particularly complex mathematical models, which inevitably feature a multitude of variables, a common and efficient strategy is consolidation through the use of vectors, where a single symbolic representation can elegantly encapsulate several distinct variables, simplifying the notation without sacrificing detail.

A priori information

When approaching a system with the ubiquitous "black box approach," the modeler deliberately limits their focus to observing only the system's stimulus/response behavior. The goal is to infer the internal workings of the (unknown) box solely from these external observations. The typical visual representation of such a black box system is a data flow diagram with the enigmatic box at its center.

Mathematical modeling problems are frequently categorized into either black box or white box models, a distinction drawn from the quantity of a priori information available about the system's internal mechanisms. A black-box model is, quite literally, a system about which virtually no prior information regarding its internal structure or governing principles is available. One is left to infer everything from observed behavior. Conversely, a white-box model (sometimes charmingly referred to as a glass box or clear box model) represents a system where all—or at least all necessary—information regarding its internal workings is readily at hand.

In practice, the vast majority of real-world systems reside somewhere on the continuum between these two idealized extremes. This conceptual framework, therefore, serves less as a rigid classification and more as an intuitive guide, helping modelers decide which methodological approach is most appropriate for their particular challenge.

Generally speaking, the more a priori information one can judiciously incorporate into a model, the greater its potential accuracy and predictive power. Hence, white-box models are typically considered "easier" to construct and validate, under the optimistic assumption that if you've correctly utilized all the available information, the model should, in theory, behave correctly. This prior information often manifests as knowledge about the functional forms relating different variables. For instance, if one were to construct a model describing how a medicine metabolizes within a human system, there's a well-established understanding that the concentration of the medicine in the blood typically follows an exponentially decaying function. However, even with this fundamental insight, several crucial parameters remain unknown: how rapidly does the medicine decay, and what was its initial concentration in the blood? This scenario, therefore, does not constitute a perfectly white-box model; these unknown parameters must be estimated through some empirical means before the model can be effectively utilized.

In the challenging domain of black-box models, the task expands to estimating not only the numerical parameters within functions but also the very functional form of the relationships between variables themselves. Without any a priori information, one is often compelled to employ functions that are as general as possible, aiming to encompass a wide spectrum of potential underlying models. A frequently employed approach for black-box modeling involves artificial neural networks, which are designed to make minimal assumptions about the incoming data, allowing them to learn complex patterns directly.

Alternatively, the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms, developed as a cornerstone of nonlinear system identification 6, offer a different paradigm. These algorithms can be systematically used to select the most relevant model terms, determine the optimal model structure, and estimate unknown parameters even in the presence of correlated and nonlinear noise. The distinct advantage of NARMAX models, when compared to the somewhat opaque nature of neural networks, is that NARMAX produces models that can be explicitly written down, interpreted, and directly related back to the underlying physical or biological process. Neural networks, while powerful approximators, often yield a "black box" solution where the internal logic remains largely inscrutable.

Subjective information

Occasionally, it proves remarkably beneficial, or perhaps just pragmatically necessary, to infuse a mathematical model with elements of subjective information. This can stem from various sources: the subtle promptings of intuition, the accumulated wisdom of experience, the distilled insights of expert opinion, or even, at times, the sheer convenience of a particular mathematical form. Bayesian statistics provides a robust theoretical framework for rigorously incorporating such subjectivity into an otherwise objective analysis. In this approach, one begins by specifying a prior probability distribution—a distribution that can, and often does, reflect subjective beliefs or existing knowledge—and then systematically updates this distribution based on newly acquired empirical data. It's a formal way of saying, "Here's what I think, now let the data tell me if I'm wrong."

Consider a scenario where such an approach becomes indispensable: an experimenter slightly bends a coin and then tosses it just once, noting whether it lands heads or tails. The task then is to predict the probability of the next flip landing heads. After the coin has been bent, its true probability of coming up heads is an unknown quantity, no longer a simple 0.5. In this situation, the experimenter would be compelled to make a decision—perhaps informed by a careful visual inspection of the coin's new shape—about what prior distribution to employ. The judicious incorporation of such subjective information, reflecting an informed initial belief, could be critically important in arriving at an accurate and useful estimate of the true probability.

Complexity

The issue of model complexity is a tightrope walk, inherently involving a perpetual trade-off between the seductive allure of simplicity and the often-elusive goal of accuracy. Occam's razor, that venerable principle of parsimony, is particularly pertinent to the art of modeling. Its essential dictate is deceptively simple: among competing models that possess roughly equivalent predictive power, the simplest one is invariably the most desirable.

While adding layers of complexity often seems to promise improved realism—a more faithful mirror to the messy reality—it frequently comes at a steep price. Such models can become agonizingly difficult to comprehend and analyze, turning into intellectual Gordian knots. Moreover, increased complexity can introduce significant computational challenges, including the dreaded specter of numerical instability, where even tiny errors propagate wildly, rendering results meaningless. Thomas Kuhn, in his insightful observations on the nature of scientific progress, argued that explanations in science tend to spiral into ever-increasing complexity before a revolutionary paradigm shift arrives, offering a radical and often elegant simplification 7. It's as if the universe occasionally rewards those who dare to simplify.

Take, for instance, the ambitious task of modeling the flight dynamics of an aircraft. One could theoretically embed every single mechanical component, down to the last rivet, into the model, thereby approaching an almost perfect white-box representation of the system. However, the sheer computational cost associated with integrating such an enormous quantity of detail would effectively render the model unusable for any practical purpose. Furthermore, the inherent uncertainty would paradoxically increase due to an overly complex system, as each individual part introduces its own minuscule amount of variance and potential error into the overall model.

It is, therefore, almost always appropriate—indeed, often imperative—to make judicious approximations to pare down the model to a sensible, manageable size. Engineers, ever the pragmatists, frequently accept certain approximations in exchange for a more robust and simpler model that can actually be used. Consider Isaac Newton's classical mechanics: it is, undeniably, an approximated model of the true underlying reality of the universe. Yet, Newton's model remains remarkably sufficient for the vast majority of ordinary-life situations, at least as long as particle speeds remain comfortably below the speed of light and our focus is solely on macro-particles. It's a stark reminder that "better accuracy" does not automatically equate to a "better model" in all contexts. Statistical models, in particular, are notoriously susceptible to overfitting, a perilous state where a model becomes so precisely tailored to the specific training data that it loses its fundamental ability to generalize effectively to new, unseen events. It's like memorizing every answer to a test but failing to understand the underlying concepts.

Training, tuning, and fitting

Any mathematical model that isn't a pristine, pure white-box—meaning it contains some inherent uncertainties or unknown aspects—will invariably possess certain parameters that can be adjusted or optimized to fit the model to the real-world system it is designed to describe. This process of parameter optimization takes on different nomenclature depending on the modeling paradigm.

If the modeling endeavor is undertaken using an artificial neural network or other advanced machine learning techniques, the process of optimizing these internal parameters is typically referred to as training. Simultaneously, the optimization of the model's hyperparameters (parameters that control the learning process itself, rather than being learned directly from data) is termed tuning, and it frequently employs sophisticated techniques like cross-validation to ensure robustness 8. In the more conventional realm of modeling, where explicitly defined mathematical functions are used, these parameters are often determined through the venerable technique of curve fitting. While the terminology may differ, the underlying goal remains consistent: to align the abstract model as closely as possible with the observed behavior of the concrete system, a perpetual dance between theory and reality.

Evaluation and assessment

A critically important, and often brutally honest, phase of the entire modeling process is the rigorous evaluation of whether a given mathematical model truly describes a system with sufficient accuracy. This question, deceptively simple on the surface, can be remarkably challenging to answer definitively, as it encompasses several distinct, yet interconnected, types of evaluation. It's where the rubber meets the road, and sometimes, the model just skids off.

Prediction of empirical data

Typically, the most straightforward aspect of model evaluation involves checking how well a model predicts experimental measurements or other empirical data that were not utilized during the model's development phase. In models that incorporate adjustable parameters, a common and highly effective strategy is to partition the available data into two distinct, non-overlapping subsets: a training data set and a verification (or test) data set. The training data are exclusively used to estimate and optimize the model's parameters. An truly accurate and robust model, one that has genuinely captured the underlying dynamics, will then closely match the verification data, despite the fact that these data points played no part in setting the model's internal parameters. This crucial practice, designed to assess a model's generalization capability, is known as cross-validation in the field of statistics.

Defining a quantitative metric to measure the "distance" or discrepancy between the observed data and the data predicted by the model is an indispensable tool for objectively assessing the model's fit. In statistics, decision theory, and certain economic models, a loss function serves a similar, critical role, quantifying the penalty for incorrect predictions. While evaluating the appropriateness of a model's parameters is generally quite manageable, assessing the validity of the model's general mathematical form—its fundamental structure—can prove significantly more difficult. In broad strokes, a more extensive array of mathematical tools has been developed to test the fit of statistical models compared to models primarily involving differential equations. Techniques drawn from nonparametric statistics can sometimes be employed to evaluate how well observed data conforms to a known distribution, or to construct a general model that makes only minimal, necessary assumptions about its own mathematical form.

Scope of the model

Assessing the scope of a model—that is, precisely determining the range of situations and conditions to which the model can legitimately be applied—is often less straightforward than merely checking its predictive accuracy. If a model was meticulously constructed and validated based on a specific set of data, the critical question then becomes: for which other systems or situations can this known data be considered "typical" or representative?

The inquiry into whether the model accurately describes the properties of the system between the observed data points is termed interpolation. Even more challenging, and often more revealing, is the same question posed for events or data points that lie outside the range of the originally observed data; this is known as extrapolation. The latter is where models often falter, revealing their inherent limitations.

As a classic illustration of the typical limitations inherent in a model's scope, consider the enduring legacy of Newtonian classical mechanics. Newton, working in his era, conducted his seminal measurements without the benefit of advanced scientific equipment. Consequently, he was unable to measure the properties of particles traveling at speeds approaching the speed of light. Similarly, his observations were focused on macro particles—planets, apples, and the like—not the elusive movements of individual molecules or subatomic particles. It should, therefore, come as no surprise that his remarkably successful model does not extrapolate well into these extreme domains. Despite these limitations, his model remains perfectly sufficient for the vast majority of ordinary-life physics applications, a testament to its practical utility within its defined scope. The lesson, as always, is that better accuracy within a specific domain does not necessarily confer universal applicability.

Philosophical considerations

Many types of mathematical modeling implicitly, and sometimes quite explicitly, involve claims about causality—the intricate web of cause-and-effect that underpins reality. This is typically, though not universally, true for models that articulate their relationships through differential equations, which inherently describe how changes in one variable cause changes in another.

Given that the overarching purpose of modeling is to deepen our understanding of the world, the true validity of a model doesn't solely hinge on its ability to perfectly fit empirical observations. Crucially, it also rests on its capacity to extrapolate meaningfully to situations or data points that extend beyond those originally used to construct and describe the model. This can be conceptualized as the fundamental differentiation between offering mere qualitative predictions versus providing precise quantitative predictions. Furthermore, one can reasonably argue that a model, no matter how mathematically elegant, is ultimately rendered rather worthless unless it furnishes some genuine insight that transcends what is already known or readily discernible from direct investigation of the phenomenon under scrutiny. If it merely confirms the obvious, what, precisely, was the point?

An example of such critical scrutiny can be found in the argument that certain mathematical models within optimal foraging theory often fail to offer insights that extend beyond the common-sense conclusions already derivable from basic principles of evolution and fundamental ecological tenets 9. It is also imperative to acknowledge that while mathematical modeling undeniably employs mathematical concepts and language, it is not, in itself, a distinct branch of pure mathematics. Consequently, it does not necessarily adhere strictly to the rigorous dictates of mathematical logic in the same way. Instead, mathematical modeling is typically understood as a specialized branch of some science or other technical subject, carrying with it the corresponding concepts, methodologies, and standards of argumentation specific to that field 10. It's a tool, not an end in itself, no matter how much mathematicians might wish otherwise.

Significance in the natural sciences

Mathematical models hold an unparalleled and truly profound significance within the natural sciences, particularly within the foundational discipline of physics. Physical theories, almost without exception, are articulated and formalized using the precise and unambiguous language of mathematical models. Throughout the annals of scientific history, there has been a relentless, often arduous, pursuit of ever more accurate and comprehensive mathematical models to describe the universe. Newton's laws, for instance, offer an exquisitely accurate description of countless everyday phenomena, providing a robust framework for understanding the macroscopic world. However, at certain extreme limits—specifically, when dealing with objects approaching the speed of light or probing the bizarre realm of the subatomic—the elegant simplicity of Newtonian mechanics must yield to the more intricate and counterintuitive frameworks of the theory of relativity and quantum mechanics, respectively. It's almost as if reality enjoys setting traps for our elegant equations.

In physics, it is a common and pragmatic practice to employ idealized models to simplify the inherent complexities of the real world, allowing for tractable analysis. Concepts such as massless ropes, infinitesimally small point particles, perfectly ideal gases, and the quintessential particle in a box are but a few examples among the myriad simplified models routinely utilized in physics education and research. The fundamental laws of physics are themselves often condensed into remarkably simple, yet profoundly powerful, equations, such as Newton's laws, Maxwell's equations (which elegantly describe electromagnetism), and the Schrödinger equation (the cornerstone of quantum mechanics). These foundational laws serve as the bedrock upon which more specific mathematical models of real-world situations are constructed.

Many real-world scenarios, however, are far too complex for direct analytical solutions derived from these basic laws. Consequently, they are often modeled approximately using computational methods, where a model that is computationally feasible to execute is derived either directly from the basic laws or from further approximate models built upon those fundamental principles. For example, molecules, with their intricate electron configurations, can be modeled using sophisticated molecular orbital models, which represent approximate solutions to the formidable Schrödinger equation. In the realm of engineering, physical models are frequently constructed and analyzed through powerful mathematical techniques such as finite element analysis, which discretizes continuous systems into smaller, manageable elements for numerical solution.

Furthermore, different mathematical models may employ diverse geometries to describe space, and these geometries are not necessarily intended to be perfectly accurate descriptions of the actual geometry of the universe. Euclidean geometry, with its flat, intuitive space, is widely employed in classical physics. However, the revolutionary theories of special relativity and general relativity stand as prime examples of frameworks that necessitate and successfully utilize non-Euclidean geometries to accurately describe the curvature of spacetime.

Some applications

Often, when engineers undertake the task of analyzing a system that is destined for control or optimization, their initial and crucial step involves the development of a mathematical model. In the analytical phase, engineers might construct a descriptive model of the system, essentially formulating a precise hypothesis about its operational mechanisms and internal workings. Alternatively, they might endeavor to estimate the potential impact of an unforeseeable event on the system's performance. Similarly, in the context of actively controlling a system, engineers can systematically test and refine various control approaches through the use of simulations, allowing them to predict outcomes and optimize strategies in a safe, virtual environment before committing to real-world implementation.

A mathematical model, at its core, typically characterizes a system by defining a specific set of variables and a corresponding set of equations that rigorously establish the relationships between these variables. These variables can be of a multitude of types: they might be real or integer numbers, binary Boolean values, or even textual strings, among others. Each variable serves to represent some quantifiable property or attribute of the system. For example, measured system outputs often manifest as electrical signals, precise timing data, incrementing counters, or records of specific event occurrences. The actual model, the beating heart of the abstract representation, is ultimately the comprehensive set of functions that precisely describe the intricate web of relations and dependencies binding the different variables together.

Examples

The ubiquity of mathematical models means they are woven into the fabric of countless disciplines and everyday activities, often without conscious recognition.

One of the most popular and foundational examples in computer science involves the mathematical models of various abstract machines. A prime illustration is the deterministic finite automaton (DFA), which is defined as a purely abstract mathematical concept. However, precisely because of the inherent deterministic nature of a DFA, it is eminently implementable in both hardware and software, serving as a powerful tool for solving a wide array of specific computational problems. For instance, consider the following DFA, denoted as M, which operates with a binary alphabet and is designed to accept any input string that contains an even number of 0s:

The state diagram for !DFA example state diagram $M=(Q,\Sigma ,\delta ,q_{0},F)$ where
- $Q=\{S_{1},S_{2}\},$ represents the set of possible states the automaton can be in.
- $\Sigma =\{0,1\},$ defines the input alphabet, meaning the automaton processes strings composed of 0s and 1s.
- $q_{0}=S_{1},$ designates the initial state of the automaton before any input is processed.
- $F=\{S_{1}\},$ specifies the set of accepting states. If the automaton finishes processing an input string in one of these states, the string is considered "accepted." and
- $\delta$ is the state-transition function, precisely defined by the following state-transition table:
0 1

$S_1$ $S_2$ $S_1$

$S_2$ $S_1$ $S_2$

In this model, the state $S_{1}$ signifies that, up to the current point in the input string, an even number of 0s has been encountered. Conversely, state $S_{2}$ indicates that an odd number of 0s has been processed. A '1' in the input stream, as evident from the table, does not alter the automaton's current state. Upon the termination of the input, the final state of the automaton will unequivocally reveal whether the input string contained an even or an odd number of 0s. If the input indeed contained an even number of 0s, $M$ will conclude its processing in state $S_{1}$ , which is an accepting state, thereby signifying that the input string is accepted. The language recognized by $M$ is the regular language precisely described by the regular expression $1^*( 0 (1*) 0 (1*) )^*$ , where "*" denotes the Kleene star operator, signifying zero or more repetitions of the preceding element (e.g., $1^*$ denotes any non-negative number, including zero, of '1' symbols). It's a rather elegant way to impose order on a stream of binary chaos.
Many routine daily activities, often carried out without a second thought, are, in essence, implicit uses of mathematical models. Consider a geographical map projection, which takes a complex, three-dimensional region of the Earth's curved surface and renders it onto a smaller, two-dimensional plane. This projection is a model, albeit a simplified one, that serves a multitude of practical purposes, such as meticulously planning travel routes 11.
Another remarkably simple yet profoundly useful activity is predicting the current position of a vehicle. This can be achieved by utilizing its initial position, its direction of travel, and its speed, applying the fundamental equation that distance traveled is simply the product of time and speed. When this method is employed with greater formality and precision, it is known as dead reckoning. What's fascinating is that this form of mathematical modeling doesn't necessarily demand formal mathematical training; even animals have been demonstrably shown to employ dead reckoning for navigation 12 13. Perhaps even the simplest minds grasp the elegance of $d=vt$ .
Population Growth: The study of how populations change over time is fertile ground for mathematical modeling. A simple, though admittedly approximate, model of population growth is the Malthusian growth model, which posits exponential growth under ideal conditions. A slightly more realistic and widely adopted model for population dynamics is the logistic function, along with its various extensions, which accounts for limiting factors and carrying capacity.
Model of a particle in a potential-field: In this model, we idealistically conceive of a particle as a mere point of mass, whose trajectory through space is meticulously described by a function that yields its spatial coordinates as a function of time. The influence of the potential field is given by a scalar function $V\!:\mathbb {R} ^{3}\!\to \mathbb {R}$ , and the particle's trajectory, represented by the vector function $\mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},$ is the solution to the following differential equation:

$-{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,$

which can be more compactly expressed using vector calculus as:

$m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].$

It's crucial to acknowledge that this model makes a significant idealization: it assumes the particle is a point mass. This assumption is, of course, known to be factually incorrect in a great many real-world applications where this model is nevertheless successfully employed; for instance, it serves as a remarkably effective model for planetary motion, despite planets being decidedly non-point masses.
Model of rational behavior for a consumer: This model, a cornerstone of microeconomics, proceeds from the rather optimistic assumption that a consumer is confronted with a choice among $n$ distinct commodities, labeled $1,2,\dots ,n$ , each possessing a specific market price, $p_{1},p_{2},\dots ,p_{n}.$ The consumer is further assumed to possess an ordinal utility function $U$ —ordinal in the sense that only the direction (sign) of differences between two utility levels, rather than their absolute magnitudes, holds meaning. This utility function is dependent on the quantities of commodities $x_{1},x_{2},\dots ,x_{n}$ consumed. The model then asserts that the consumer operates within a finite budget $M$ , which is allocated to purchase a vector of commodities $x_{1},x_{2},\dots ,x_{n}$ in such a manner as to maximize their utility $U(x_{1},x_{2},\dots ,x_{n}).$

The problem of rational consumer behavior in this model thus transforms into a classic mathematical optimization problem:

$\max \,U(x_{1},x_{2},\ldots ,x_{n})$

subject to the following constraints:

$\sum _{i=1}^{n}p_{i}x_{i}\leq M,$ (the total expenditure must not exceed the budget) $x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots ,n.$ (the quantity of each commodity consumed cannot be negative).

This model, despite its simplifying assumptions about human behavior, has found extensive application across a wide spectrum of economic contexts. For example, it is instrumental in general equilibrium theory for demonstrating the existence and Pareto efficiency of economic equilibria.
Neighbour-sensing model: This fascinating biological model endeavors to explain the organized formation of mushrooms from an initially chaotic and diffuse fungal network, illustrating how localized interactions can lead to macroscopic patterns.
In computer science, mathematical models are frequently employed to simulate the intricate behavior and performance of complex computer networks, allowing for analysis, prediction, and optimization without physical construction.
In mechanics, mathematical models are indispensable tools used to meticulously analyze and predict the movement and trajectory of objects, such as a rocket model, under various forces and conditions.

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