Mathematical modeling
Mathematical modeling is a research method in which a real object, process, or phenomenon is replaced by its mathematical model, allowing for the analysis of a system's behavior using mathematical methods.
Essence of Mathematical Modeling
Mathematical modeling includes:
- formalization of the essential characteristics of the object under study;
- construction of a mathematical model that expresses the relationships between parameters;
- analysis of the model using analytical, numerical, or simulation methods;
- interpretation of the results in the context of the real system.
A model is always a simplification of reality, purposefully reflecting only those aspects that are important for the stated research or design goals.
Goals of Mathematical Modeling
Mathematical models are developed for:
- describing the structure, properties, and functioning of systems;
- explaining observed phenomena by identifying patterns;
- predicting the future behavior of systems under given influences;
- optimizing processes and control systems;
- conducting virtual experiments that are inaccessible or undesirable in reality.
Classification of Mathematical Models
- By method of description: Deterministic and stochastic (probabilistic).
- By time characteristic: Static and dynamic.
- By variable characteristic: Discrete and continuous.
- By level of detail: Macroscopic, microscopic, mesoscopic.
- By mathematical framework: Analytical, numerical, simulation-based.
- By purpose: Descriptive, predictive, optimization, simulation.
- By structure: Linear and nonlinear.
- By construction approach: Phenomenological (empirical) and mechanistic (theoretical).
Stages of Mathematical Modeling
In general, the stages of mathematical modeling are as follows:
- Problem Formulation:
- Defining the modeling goal (what do we want to know or do?).
- Describing the object or process and its boundaries.
- Identifying the key factors and characteristics to be considered.
- Model Construction (or Selection):
- Formalization: Translating the description of the object and its relationships into the language of mathematics (equations, functions, logical rules, algorithms, etc.).
- Introducing assumptions and simplifications to isolate the main aspects.
- Defining the model's parameters and their relationships.
- Model Analysis:
- Solving the mathematical problem (analytically, numerically, or through simulation).
- Conducting computational experiments to study the model's behavior under different conditions.
- Model Adequacy Check (Validation):
- Comparing the modeling results with real data (experimental, observational) or known facts.
- Assessing how well the model reflects reality for the stated purpose.
- Interpretation and Application of Results:
- Analyzing the data obtained from the model.
- Translating the results back into the language of the original problem.
- Formulating conclusions, predictions, developing recommendations, or making decisions based on the model.
This process is often iterative. Unsatisfactory validation or interpretation results may require returning to previous stages (refining the problem formulation, changing assumptions, modifying the model).
A Principled Approach
A principled approach, based on the work of Clive Dym (Principles of Mathematical Modeling), suggests viewing modeling as a sequence of answers to key questions:
- Why?: What is the need for the model? It is necessary to clearly define the modeling goal and the problem the model is intended to help solve.
- Find?: What do we want to learn? What specific outputs, characteristics, or information should the model provide to achieve the goal?
- Given?: What do we know? What information, data (experimental, statistical), knowledge about the system, and available resources do we already have?
- Assume?: What assumptions are we making? Defining the simplifications, idealizations, hypotheses about the system's behavior, and the limits of the model's applicability. This step is critical as it determines the model's adequacy.
- How?: How does the system function? Identifying the fundamental laws (physical, chemical, biological, economic, etc.), mechanisms, and interconnections that govern the behavior of the object being modeled.
- Predict?: What will the model predict? Formulating the mathematical equations, inequalities, logical rules, or algorithms that form the core of the model, and defining the calculations to be performed.
- Valid?: How well do the model's predictions correspond to reality? Comparing the modeling results with real data or known facts to check the model's adequacy for the stated goals.
- Verified?: Is the model useful for achieving the original goal (Why?)? Do the obtained results and the model's accuracy satisfy the initial need? (Verification—checking the correctness of mathematical calculations and software implementation—may also be performed at this stage).
- Improve?: Can and should the model be improved? Identifying parameters that need refinement, assumptions that can be relaxed, or unconsidered aspects that should be added for greater accuracy or broader application.
- Use?: How to apply the results? Interpreting the model's predictions and conclusions to make decisions, gain new knowledge, forecast, optimize, or control the system.
Forward and Inverse Problems in Mathematical Modeling
Traditionally, two main classes of problems associated with mathematical models are distinguished:
- The forward problem involves studying a model with a predetermined structure and known parameters to obtain information about the object's behavior.
- The inverse problem involves selecting a specific model or determining its parameters based on available experimental or empirical data.
The Forward Problem
The goal of the forward problem is to answer questions about a system's response to external influences or to determine its characteristics under various conditions, based on its known properties. Key aspects of the forward problem's goal include:
- Investigating the object's behavior based on a given mathematical model with a known structure and parameters.
- Obtaining quantitative or qualitative characteristics of the system: for example, determining stresses, temperature fields, dynamic responses to loads, etc.
- Predicting the object's state under various external influences (loads, changes in environmental conditions, control actions).
- Analyzing the stability and reliability of systems, and determining their operational limits.
- Testing hypotheses about the object's behavior formulated based on the model.
- Optimizing control processes by calculating the model's responses to control inputs.
- Assessing the sensitivity of solutions to changes in initial conditions and model parameters.
The Inverse Problem
The goal of the inverse problem is to determine the structure of a model or its parameters based on available data about the real system's behavior.
Key aspects of the inverse problem's goal include:
- Finding unknown model parameters (e.g., coefficients of elasticity, thermal conductivity, resistance, etc.).
- Identifying the hidden structure of processes based on observed output data.
- Constructing or correcting a mathematical model so that its behavior is consistent with experimental or empirical data.
- Developing adequate methods for processing and interpreting observational and experimental data.
Verification and Validation
After a mathematical model has been constructed and implemented in software or as an algorithm, a critical step is to assess its correctness and applicability. This is done through two interconnected but distinct processes: verification and validation.
- Verification:
- Question: Are we building the model right?
- Meaning: Checking that the software implementation or computational algorithm of the model accurately corresponds to its mathematical formulation and conceptual description. In other words, verification ensures that the equations are solved correctly and the algorithm works without errors according to its intended logic.
- Methods: Code analysis and review, testing against known analytical solutions (if they exist), comparison with results from other verified programs, checking the numerical stability and convergence of algorithms.
- Validation:
- Question: Are we building the right model?
- Meaning: Determining the degree to which the model corresponds to the real object, process, or phenomenon it is intended to describe, in relation to the modeling goals. Validation answers the question of how well the model reflects the aspects of reality we are interested in.
- Methods: Comparing modeling results with experimental data, observational data from the real system, statistical data, known facts, or results from other well-established models. Assessing the model's sensitivity to parameter changes.
Key Distinction:
- Verification compares the model's implementation with its specification (mathematical description).
- Validation compares the model (as a whole) with reality.
It is possible to have a mathematically correct model that is inadequate for the real process (verified but not validated), or a model intended to be adequate but implemented with errors (not verified).
The processes of verification and validation are an integral part of mathematical modeling. They build confidence in the modeling results and help to assess the model's limits of applicability. Without them, using a model for prediction, optimization, or decision-making can be incorrect.
Limitations and Assumptions
Any mathematical model is an abstraction—a purposeful simplification of the reality under study. It is based on a set of assumptions and, like any map, is not the territory itself but merely a description of it for specific purposes. This imposes the following limitations:
- Incompleteness: The model considers only the factors and relationships deemed essential (from the perspective of the research goal and the assumptions made), deliberately ignoring other details to simplify analysis.
- Dependence on Assumptions: The correctness, accuracy, and scope of applicability of the model directly depend on the validity of the assumptions made. Incorrect or violated assumptions lead to inaccurate or wrong results.
- Limited Scope of Applicability: The model adequately describes reality only under specific conditions (consistent with the assumptions) and for solving the specific range of problems for which it was created and tested. Extrapolation beyond these boundaries is incorrect.
- Inaccuracy: Due to simplifications, the results of modeling always have some degree of error compared to the real behavior of the object.
- Need for Validation: Since a model is a simplification, its adequacy always requires verification (validation) based on real (experimental or observational) data.
- Sensitivity: Modeling results can be sensitive to changes in input data, model parameters, and the assumptions made, which often requires a sensitivity analysis.
Application Examples
- In physics: modeling heat conduction, fluid dynamics, electromagnetic processes.
- In engineering: calculating the strength of structures, optimizing technological processes.
- In biology: modeling population dynamics and the spread of diseases.
- In economics: building macroeconomic models and optimal control models.
- In sociology: modeling migration processes and social dynamics.
Literature
- Introduction to Mathematical Modeling / Ed. P.V. Trusov. — Moscow: Universitetskaya kniga, Logos, 2007.
- Mathematical Modeling. Ideas. Methods. Examples. Moiseev N.N. — Moscow: Nauka, 1981.
- Mathematical Modeling. Galanin M.P., Galanina E.M., Sergeev A.V., Shalaeva A.K. — Moscow: LKI, 2022.
- Principles of Mathematical Modeling. Dym, C.L. (2004).