I. Introduction
This article is mainly aimed at students who have never understood mathematical modeling. It helps students how to quickly get started with mathematical modeling and win awards in various competitions as soon as possible, or write excellent mathematical modeling papers.
In this article, I will explain how to get started with mathematical modeling starting from what mathematical modeling is, the application fields of mathematical modeling, the basic steps of mathematical modeling, and the techniques and tools of mathematical modeling. Among them, I hope you can focus on learning and understanding the basic steps, techniques, and tools of mathematical modeling. Because in actual competitions, the background of specific problems in mathematical modeling will be given. Of course, if you want to know which type of mathematical modeling background corresponds to what methods and ideas, you can read my previous articles. The link is as follows: Mathematical Modeling |
Problems Corresponding to Common Algorithms_Shuofangniao's Blog-CSDN Blog 
https://blog.csdn.net/as12138/article/details/128378585?spm=1001.2014.3001.5501
2. What is mathematical modeling?
Mathematical modeling is the process of transforming real-life problems into mathematical models. It is a multidisciplinary field that requires knowledge from multiple disciplines such as mathematics, statistics, computer science, physics, and engineering. Through mathematical modeling, real-life problems can be quantitatively analyzed and suitable solutions can be designed. Mathematical modeling is widely used in various fields, such as economy, environmental protection, urban planning, medicine, etc.
Mathematical modeling is the process of abstracting problems in the real world and using mathematical tools and methods to describe, analyze and solve these problems. It is the intersection of mathematics and practical applications, aiming to understand the nature of the problem, predict and optimize the behavior of the system, and provide decision support by establishing mathematical models.
3. Application areas of mathematical modeling
① Mathematical modeling cases in the engineering field
Mathematical modeling plays an important role in the field of engineering. For example, construction engineers can use mathematical models to predict and optimize a building's structure and performance to ensure its safety and reliability. The planning and operation of power systems can be aided by mathematical models to achieve optimal energy distribution and efficiency. Production scheduling and optimization problems in manufacturing are often solved through mathematical modeling to improve production efficiency and resource utilization.
② Mathematical modeling cases in the fields of economics and finance
In economics and finance, mathematical modeling is widely used in risk management, portfolio optimization, market forecasting and decision analysis. Financial institutions can use mathematical models to evaluate investment risks and returns and formulate reasonable asset allocation strategies. Economists can use mathematical models to study issues such as market competition, price decisions, and economic policy.
③ Mathematical modeling cases in the fields of medicine and biological sciences
Mathematical modeling plays an important role in the fields of medicine and biological sciences. For example, biologists can use mathematical models to study the dynamics and stability of biological systems and analyze gene regulatory networks and cell signaling mechanisms. Disease transmission and epidemic models in the medical field can help researchers predict and control the spread paths and prevention strategies of infectious diseases. In addition, the processing and analysis of medical images are often implemented with the help of mathematical models.
③ Mathematical modeling cases in the fields of natural science and environment
Many problems in natural sciences and environmental science research can be solved through mathematical modeling. For example, astrophysicists can use mathematical models to describe the evolution of galaxies and the orbital motion of planets in the universe. Meteorologists use mathematical models to predict and simulate climate change, weather patterns and storm systems. Ecologists can use mathematical models to study ecosystem stability, species distribution, and biodiversity.
The application fields of mathematical modeling go far beyond this. It has extensive applications in transportation planning, energy management, social sciences, information technology and other fields. Through mathematical modeling, we can better understand the operating rules of complex systems, predict and optimize system behavior, and provide scientific solutions to practical problems.
4. Basic steps of mathematical modeling
The process of mathematical modeling can be simply summarized into the following steps: defining the problem, collecting data, building a model, solving analysis and interpreting the results.
Step1: Define the problem and establish goals
The first step in mathematical modeling is to clarify the problem and set the goals of the modeling. We need to carefully analyze the background of the problem and the core problem to be solved, and clarify what we hope to achieve through modeling. This helps us better select modeling approaches and identify factors and variables to consider.
Step2: Collect and organize relevant data
Before establishing a mathematical model, we need to collect relevant data, organize and analyze it. Data can come from experimental observations, questionnaires, literature research, etc. The data collected should be representative and accurate, and should be closely related to the context and objectives of the problem.
Step3: Establish mathematical model
On the basis of collecting and analyzing data, we begin to build mathematical models. Mathematical models are abstractions and simplifications of practical problems. They can take the form of mathematical equations, statistical relationships, optimization models, etc. Building a mathematical model requires selecting appropriate model forms and assumptions based on the characteristics of the problem, and ensuring that the model can accurately describe the essence of the problem.
Step4: Model solving and analysis
After establishing the mathematical model, we need to solve and analyze the model. This requires the application of mathematical reasoning, computational methods and tools to solve mathematical models. The solution process may involve numerical calculations, symbolic calculations, optimization algorithms, etc. Through model solving, we can obtain information such as answers to questions, calculation results, and system behavior.
Step5: Interpretation and application of results
Finally, we interpret and apply the solved results. This means translating the results of mathematical models into solutions to real problems or decision recommendations. Interpreting results requires converting mathematical language into understandable language and relating the results to a practical context so that they can inform decision-makers or stakeholders.
The process of mathematical modeling is an iterative process. In practical applications, we may need to continuously adjust the parameters and assumptions of the model, re-collect and analyze data, and perform model solving and analysis to obtain more accurate and feasible results. By continuously refining and optimizing mathematical models, we are able to better understand problems, predict system behavior and provide effective solutions.
5. Mathematical modeling techniques and tools
(1) Common mathematical methods and techniques
① Analysis and reasoning: In the mathematical modeling process, logical reasoning and mathematical analysis methods need to be used to understand the problem, derive the model and analyze the results.
② Calculus and difference equations: Calculus and difference equations are important tools for establishing dynamic models and describing changes. They can be used to describe the change patterns and stability of the system.
③ Statistics and probability: Statistical analysis and probability models can help analyze the distribution characteristics of data, discover correlations and perform parameter estimation to support modeling and decision analysis.
(2) Application of mathematical software and programming languages
① MATLAB: MATLAB is a powerful mathematical modeling and calculation tool that provides a wealth of mathematical functions and visualization tools, suitable for solving and analyzing various mathematical models.
② R language: R language is a popular statistical modeling and data analysis tool. It provides a wealth of data processing and statistical analysis functions that can be used to process large-scale data and implement complex models.
③ Python: Python is a general-purpose programming language with a wide range of mathematical computing libraries and scientific computing tools, such as NumPy and SciPy, suitable for various mathematical modeling and data analysis tasks.
(3) Modeling and analysis tools
① Optimization tools: Optimization tools can help us solve optimization problems, such as linear programming, nonlinear programming and integer programming. Commonly used optimization software packages include Gurobi, CPLEX and AMPL, etc.
② Discrete event simulation: Discrete event simulation software can simulate and analyze complex discrete event systems, such as queuing systems, supply chain networks, etc., such as Arena and Simio.
③ Data mining tools: Data mining tools can help us extract useful information and patterns from large amounts of data. Commonly used data mining software packages include Weka, RapidMiner, and Python's Scikit-learn.
It should be emphasized that the above tools and methods are collected from the Internet, are not comprehensive, and are not necessarily the optimal problem-solving ideas in a certain problem environment. At the same time, although the author has participated in many mathematical modeling competitions, there are still many methods that he has not mastered or even heard of.
Therefore, it is recommended that when preparing for mathematical modeling, you give priority to preparing the most commonly used environments. In terms of competition time, even if you have not been exposed to certain methods and backgrounds before the competition, you can still learn them during the competition.
Commonly used problem backgrounds and algorithms are as follows:
Mathematical modeling | Problems corresponding to commonly used algorithms_Shuofangniao's blog-CSDN blog https://blog.csdn.net/as12138/article/details/128378585?spm=1001.2014.3001.5501
6. Conclusion
Mathematical modeling is a process of abstracting real-life problems and using mathematical tools and methods to analyze and solve them. By building mathematical models, we can deeply understand the nature of the problem, predict and optimize the behavior of the system, and provide decision support and solutions for practical applications.
The basic steps of mathematical modeling include problem definition, data collection, model establishment, solution analysis, and result interpretation. During the modeling process, we can use a range of techniques and tools, such as mathematical methods, software and programming languages, to support model development, solution and analysis.
In this article, please briefly understand the definition of mathematical modeling, roughly know the application fields of mathematical modeling, become familiar with the basic steps of mathematical modeling, and selectively master the techniques and tools of mathematical modeling . I hope this article can provide some help to students who want to participate in or learn mathematical modeling.