Mastering Mathematical Modeling: Key Requirements and Step‑by‑Step Process

General Requirements for Building a Mathematical Model

In general, building a mathematical model should meet the following requirements:

(1) Sufficient accuracy – capture essential relationships and laws while discarding non‑essential aspects. (2) Simplicity – the model must be tractable; overly complex models are unsolvable or hard to solve. (3) Adequate basis – rely on scientific and objective laws when formulating formulas, charts, or algorithms. (4) Prefer standard forms. (5) The represented system should be controllable and testable, facilitating verification and modification.

General Steps for Building a Mathematical Model

Real‑world problems are often complex with many influencing factors. Trying to include every factor makes a model impractical; therefore a balance between simplicity and representativeness is needed. The following steps provide a general framework, but practitioners should adapt them to specific cases.

(1) Model Preparation

Understand the background of the problem, clarify the modeling purpose, and gather relevant information such as data and literature. Conduct thorough investigations, consult experts when needed, and separate important variables from peripheral ones. Describe the problem in natural language and preliminarily identify variables and their relationships.

(2) Model Assumptions

Based on the object’s characteristics and the modeling goal, and using the collected data, propose hypotheses that simplify the problem. Express these assumptions in precise mathematical language. Reasonable assumptions are crucial; overly simplistic or overly detailed assumptions can lead to model failure or excessive complexity.

(3) Model Construction

Using the assumptions, apply appropriate mathematical tools to represent relationships among variables, creating formulas, tables, or diagrams. Choose tools that match the problem’s nature, the modeling objectives, and the modeler’s expertise, favoring simplicity to enhance understandability and usability.

(4) Model Solving

Solve the model with suitable techniques—analytical solutions, graphical methods, logical reasoning, theorem proving, stability analysis, etc.—requiring relevant mathematical knowledge and computational skills.

(5) Model Analysis

Analyze the solution mathematically, interpreting variable dependencies, predicting outcomes, or deriving optimal decisions and controls based on the problem’s nature.

(6) Model Verification

Translate the analytical results back to the real system and compare them with actual data or observations to assess correctness and applicability. Successful verification shows the model can explain known phenomena and predict new ones, as exemplified by Newton’s law of universal gravitation.

If verification reveals discrepancies, investigate the cause—often the assumptions—and revise the model accordingly. Complete reliance on common sense is discouraged because intuition may be misleading.

(7) Model Application

Applying the model constitutes an additional test. Each application should be treated as a verification step, paying close attention to the simplifications made earlier. Adjustments such as merging similar variables, treating minor variables as constants, or converting between continuous and discrete representations may be necessary to maintain fidelity.

Shen Wenxuan, Yang Qingtiao, Mathematical Modeling Attempts