Mathematical Models

- Mathematical Models 

A mathematical model is a mathematical relation that describes a real-life situation. It's an abstract description of a concrete system using mathematical concepts and language.
Mathematical models are used to solve real-life situations, such as: 

• Launching a satellite
• Predicting the arrival of the monsoon
• Controlling pollution due to vehicles

Mathematical models are used in all fields of physical science, including biology, physics, and chemistry. They are also used in engineering and some social sciences. 

The most common type of mathematical model is the equation. A given scenario can be modeled with an equation, or formula, and that equation can be used to answer different questions about the scenario. 

Mathematical models can be divided into either numerical models or analytical models.
Here are some steps for creating a mathematical model: 

• Specify the problem
• Set up a metaphor
• Formulate mathematical model
• Solve mathematical model
• Interpret solution
• Compare with reality
• Use model to explain, predict, decide, design

The goal of the modeler is to ensure that the model replicates the phenomena being modeled to an acceptable degree. The procedures followed to test the fidelity of the model in reproducing the real system it represents constitute model validation.

- AI-based Mathematical Modeling

AI-based mathematical modeling is a relatively new technology that combines deep learning algorithms with traditional methods to solve complex problems. 

Mathematics is the foundation of AI algorithms and models, allowing machines to process, analyze, and interpret large amounts of data. Concepts from linear algebra, calculus, probability theory, and statistics are essential for developing ML algorithms. 

Here are some examples of AI-based mathematical modeling:

• Wolfram Alpha: A widely recognized AI-powered computational knowledge engine that can perform a wide range of mathematical operations.
• Generative AI: A subset of ML that relies on mathematical models to generate new data instances that resemble a given set of data.
• AI chatbots: Use linear algebra in tasks such as word embeddings, which convert words into numerical vectors for analysis and understanding.

Some popular AI models include:

• Linear Regression
• Deep Neural Networks
• Logistic Regression
• Decision Trees
• Linear Discriminant Analysis
• Naive Bayes
• Support Vector Machines
• Learning Vector Quantization

[More to come ...]