The Essential Process of Mathematical Modelling | 数学模型构建的基本流程

📚 The Essential Process of Mathematical Modelling | 数学模型构建的基本流程

Mathematical modelling is the art of translating real-world situations into mathematical forms, solving the resulting problem, and interpreting the solution back in the original context. It is a core skill in A-Level Mathematics, particularly for mechanics, statistics, and decision mathematics.

数学模型构建是将现实情境转化为数学形式、求解所得问题并将结果还原到原始情境中的艺术。它是A-Level数学的核心技能，尤其在力学、统计学和决策数学中至关重要。

---

1. Understanding the Real-World Problem | 理解现实问题

Every model begins with a clear statement of the problem. You must identify what question needs answering, who needs the answer, and what level of accuracy is required. A poorly defined problem leads to a useless model.

每一个模型都始于对问题的清晰陈述。你必须明确需要回答什么问题、谁需要答案以及需要什么样的精度水平。定义不清的问题会导致模型无用。

• State the objective: Are we predicting, explaining, or optimising?

  陈述目标：我们是预测、解释还是优化？

• Identify the system boundary: What is inside the model and what is excluded?

  确定系统边界：模型内部包含什么，排除什么？

• Set success criteria: How will we know if the model is good enough?

  设定成功标准：我们如何知道模型是否足够好？

For example, when modelling the trajectory of a projectile, you first decide whether air resistance is relevant, whether Earth’s curvature matters, and what range of speeds you are considering.

例如，在模拟抛射体轨迹时，你首先要决定空气阻力是否相关，地球曲率是否重要，以及考虑什么速度范围。

---

2. Making Simplifying Assumptions | 作出简化假设

No model can capture every detail of reality. Assumptions reduce complexity while preserving the essential behaviour of the system. In A-Level mechanics, common assumptions include neglecting air resistance, treating objects as particles, and assuming smooth surfaces.

没有任何模型能捕捉现实的所有细节。假设在保留系统本质行为的同时降低复杂性。在A-Level力学中，常见假设包括忽略空气阻力、将物体视为质点以及假设表面光滑。

Assumptions should be stated explicitly and justified. They are not weaknesses; they define the domain of validity of the model. A model without assumptions is not a model at all.

假设应当明确陈述并说明理由。它们不是弱点，而是定义模型的有效范围。没有假设的模型根本不是模型。

Real world → Idealised world → Mathematical world

现实世界 → 理想化世界 → 数学世界

---

3. Identifying Variables and Parameters | 识别变量与参数

Variables are quantities that change within the model, such as displacement, velocity, or population size. Parameters are constants that characterise the system, such as gravitational acceleration g or the spring constant k.

变量是在模型内变化的量，如位移、速度或种群数量。参数是表征系统的常量，如重力加速度 g 或弹簧常量 k。

• Independent variables: usually time t or position x, chosen freely.

  自变量：通常为时间 t 或位置 x，可自由选择。

• Dependent variables: quantities that respond to changes in the independent variable.

  因变量：随自变量变化而响应的量。

• Parameters: fixed values given by the context, e.g., mass m, damping coefficient c.

  参数：由情境给定的固定值，如质量 m、阻尼系数 c。

Every variable and parameter must be defined with its unit. Mixing units is one of the most common causes of modelling errors in A-Level coursework.

每个变量和参数都必须定义其单位。混用单位是A-Level课程作业中导致建模错误最常见的原因之一。

---

4. Formulating Mathematical Relationships | 构建数学关系式

This is the heart of modelling. You connect variables and parameters using equations, inequalities, or logical rules. The choice of mathematical structure depends on the behaviour of the system.

这是建模的核心。你使用方程、不等式或逻辑规则将变量和参数连接起来。数学结构的选择取决于系统的行为。

For continuous change, differential equations are natural tools. For optimisation under constraints, linear programming is used. For counting problems, combinatorics and graph theory come into play.

对于连续变化，微分方程是自然的工具。对于约束下的优化，使用线性规划。对于计数问题，则涉及组合数学和图论。

Simple population growth: dP/dt = kP

简单人口增长模型：dP/dt = kP

where P is population size and k is the intrinsic growth rate. This linear differential equation leads to exponential growth, which may be adequate for short-time predictions but unrealistic in the long run.

其中 P 是种群数量，k 是内禀增长率。该线性微分方程导出指数增长，这对短期预测可能足够，但长期来看不切实际。

---

5. Choosing a Modelling Approach | 选择建模方法

There are three broad approaches: analytical, numerical, and statistical. Analytical methods give exact solutions; numerical methods approximate solutions using algorithms; statistical methods infer patterns from data.

建模方法大致有三类：解析法、数值法和统计法。解析法给出精确解；数值法用算法逼近解；统计法从数据中推断规律。

Approach	When to use	A-Level example
Analytical	Simple geometry, constant acceleration	SUVAT equations
Numerical	Complex ODEs, no closed form	Euler’s method
Statistical	Data available, randomness involved	Normal distribution fitting

The chosen approach determines the tools you will use: algebra, calculus, probability, or computational iteration. Sometimes a hybrid approach is best.

所选方法决定了你要使用的工具：代数、微积分、概率或计算迭代。有时混合方法是最好的。

---

6. Solving the Model | 求解模型

Once the model is formulated, you must solve it. This may involve solving equations, integrating differential equations, or running simulations. In exams, you are often given a ready-made model to solve. In coursework, you must also derive it.

模型构建完成后，你必须求解它。这可能涉及解方程、积分微分方程或运行模拟。在考试中，你通常被给予现成的模型去求解。在课程作业中，你还需要自己推导。

Key techniques include separation of variables, integrating factors, differentiation of parametric equations, and numerical methods like the trapezium rule or Euler’s method.

关键技术包括分离变量法、积分因子、参数方程的微分以及数值方法，如梯形法则或欧拉方法。

Euler’s method: yₙ₊₁ = yₙ + h · f(xₙ, yₙ)

欧拉方法：yₙ₊₁ = yₙ + h · f(xₙ, yₙ)

Always check your solution for mathematical reasonableness: does it satisfy the initial conditions? Does it have the right behaviour as t → ∞?

始终检查你的解在数学上是否合理：它是否满足初始条件？当 t → ∞ 时它是否有正确的行为？

---

7. Validating the Model | 验证模型

Validation compares the model’s predictions with real-world data or known results. If your model predicts a falling object reaches 100 m/s after 10 seconds, but measurements show 95 m/s, you may need to include air resistance.

验证是将模型的预测与现实数据或已知结果进行比较。如果你的模型预测一个下落物体在10秒后达到 100 m/s，但测量显示 95 m/s，你可能需要加入空气阻力。

• Compare predictions against actual data points.

  将预测与实际数据点进行比较。

• Check edge cases: what happens when a parameter becomes zero or infinite?

  检查边界情形：当某个参数变为零或无穷大时会发生什么？

• Perform sensitivity analysis: which parameters affect the output most strongly?

  进行敏感性分析：哪些参数对输出影响最强？

A model is never completely true; it is simply useful or not useful. If the error is within the acceptable range, the model is adequate.

模型永远不是完全真实的；它只是有用或没用的区别。如果误差在可接受范围内，模型就是足够的。

---

8. Interpreting Results and Communicating | 解释结果与沟通

Finally, translate the mathematical solution back into the context of the original problem. A differential equation solution is meaningless until you explain that “the population reaches carrying capacity after 40 years.”

最后，将数学解翻译回原始问题的情境中。微分方程的解毫无意义，除非你解释为“种群在40年后达到环境容纳量”。

Good communication includes clearly stating the units, the limitations of the model, and the confidence level of the predictions. This is what distinguishes a mathematician from a calculator.

良好的沟通包括明确说明单位、模型的局限性以及预测的置信水平。这正是区分数学家与计算器的关键。

Conclusion: “The bridge collapses at wind speed 38 m/s” not “v_c = 38 m/s”

结论：写“桥梁在风速 38 m/s 时坍塌”，而不只是写“v_c = 38 m/s”

---

9. Iterating and Refining the Model | 迭代与精化模型

Modelling is rarely a one-shot process. The first model may be too simple or too complex. The cycle of refine, re-solve, and re-validate is natural. In A-Level terms, you might start with a constant-acceleration model and then refine it to include variable acceleration.

建模很少是一次性过程。第一个模型可能过于简单或过于复杂。精化、重新求解、重新验证的循环是自然的。就A-Level而言，你可能会从恒加速度模型开始，然后精化为变加速度模型。

Each iteration should improve the model’s fit without making it unnecessarily complicated. Occam’s razor applies: choose the simplest model that meets the success criteria.

每次迭代都应改进模型的拟合度，同时避免不必要的复杂化。奥卡姆剃刀原则适用：选择满足成功标准的最简单模型。

---

10. A Worked Example: Newton’s Law of Cooling | 案例研究：牛顿冷却定律

Problem: A cup of tea at 90°C is placed in a room at 20°C. After 5 minutes, it cools to 60°C. Find the time when it reaches 40°C.

问题：一杯 90°C 的茶放在 20°C 的房间里。5分钟后冷却到 60°C。求它达到 40°C 的时间。

Step 1: Assumption: cooling rate is proportional to the temperature difference between the tea and the room.

第1步：假设：冷却速率与茶水与房间之间的温差成正比。

Step 2: Variables: T = temperature of tea, t = time in minutes. Parameter: room temperature T₀ = 20°C, constant k.

第2步：变量：T = 茶水温度，t = 时间（分钟）。参数：室温 T₀ = 20°C，常数 k。

Step 3: This gives the differential equation:

第3步：这给出微分方程：

dT/dt = -k(T – 20)

Separating variables and integrating gives T = 20 + 70e⁻ᵏᵗ. Using T(5) = 60, we find:

分离变量并积分得 T = 20 + 70e⁻ᵏᵗ。利用 T(5) = 60，我们求得：

40 = 20 + 70e⁻ᵏᵗ → e⁻ᵏᵗ = 2/7 → t = (1/k) ln(7/2)

Numerically, k = (1/5)ln(7/4) ≈ 0.1119, so t ≈ 11.2 minutes. This is the mathematical answer, but a good modeller adds: “The model ignores heat loss from evaporation and assumes uniform tea temperature, so the true value may differ slightly.”

数值上，k = (1/5)ln(7/4) ≈ 0.1119，所以 t ≈ 11.2 分钟。这是数学答案，但好的建模者还会补充：“该模型忽略蒸发散热并假设茶水温度均匀，因此真实值可能略有不同。”

---

11. Common Pitfalls in Modelling | 建模常见错误

Even experienced mathematicians make modelling mistakes. Here are the most frequent ones found in A-Level submissions.

即使是经验丰富的数学家也会犯建模错误。以下是A-Level作业中最常见的错误。

Pitfall	Why it is wrong	Fix
Overfitting	Model matches data but predicts poorly	Use fewer parameters
Unit mismatch	Numbers become meaningless	Convert all units first
Ignoring limitations	Misleads the reader	State assumption boundaries
Circular reasoning	Using the answer to prove the answer	Check the derivation logic

Always revisit your assumptions after solving. If a model gives absurd results, the assumptions — not the algebra — are usually the first place to look.

求解后务必重新审视你的假设。如果模型给出荒谬的结果，首先应检查的通常是假设，而不是代数推导。

---

12. The Modelling Cycle in Exams | 考试中的建模循环

In A-Level mathematics, examiners are not just looking for the final answer. They award marks for each stage of the modelling process: defining variables, forming equations, solving correctly, and interpreting the result.

在A-Level数学中，考官不只是看最终答案。他们为建模过程的每个阶段打分：定义变量、建立方程、正确求解和解释结果。

A complete answer to a modelling question should show the FULL cycle: understanding, assumptions, formulation, solution, validation, interpretation. Even a simple kinematics problem benefits from this structured thinking.

建模题的完整答案应展示完整的循环：理解、假设、构建、求解、验证、解释。即使是一个简单的运动学问题也能从这种结构化思维中受益。

Practise this cycle with past paper questions. You will find that your solutions become clearer, and you are less likely to make careless errors.

用历年真题练习这个循环。你会发现解题思路变得更清晰，也更不容易犯粗心错误。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）