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Learning Artificial Intelligence (AI) for Mechanical Engineers is Easier than You Think!

“Mathematics is the language with which God has written the universe.” -Galileo Galilei

In mechanical engineering, my journey as an undergraduate student was marked by a profound focus on 3D CAD design and simulations. Never once I felt inclined towards the application of AI for mechanical engineers in their field. Throughout my four-year program, I became proficient in using essential mechanical software tools such as Ansys, Abaqus, and SolidWorks. These tools were the backbone of our mechanical engineering curriculum, helping us explore the intricacies of designing and simulating complex systems. Coding and programming, on the other hand, often seemed like distant lands reserved exclusively for computer engineers and software developers. Little did I know then, the vital role mathematics, particularly calculus, linear algebra, numerical methods, probability, and statistics, would play in bridging this gap and paving the way for a deeper understanding of artificial intelligence (AI) for mechanical engineers.  

Understanding Artificial Intelligence (AI):

AI, often portrayed as an enigmatic field of futuristic robots and sci-fi fantasies, is, at its core, rooted in mathematics and algorithms. It’s about matrices and tensors being multiplied, iterated, and updated to mimic human-like intelligence. While AI follows well-defined algorithms to complete tasks, it’s essential to understand that these algorithms aren’t just recipes you translate into programming languages like C or Python without thought. They are deeply rooted in mathematical concepts, logic, and understanding.

For mechanical engineers, delving into machine learning and programming brings math to life in ways we might not have initially anticipated during our first university years. Clearly, you can’t learn too much math regarding AI. Our strong foundation in mathematical concepts provides a distinct advantage, making it easier for us to grasp the science behind these algorithms.

The core of AI isn’t science fiction or magic; it’s mathematics. Math is often referred to as the language of the universe, and AI employs math to simulate human intelligence in solving tasks that the human brain doesn’t naturally compute with algorithms. The idea of thinking machines and the ability to replicate human behavior are made possible by applying mathematical concepts.

Linear Algebra:

Linear algebra is a branch of mathematics that deals with vectors, matrices, and, more generally, vector spaces and linear transformations. It forms the foundation for various fields of science and engineering, including physics, computer science, machine learning, and, of course, mathematics itself.

Here are some key concepts and components of linear algebra:

 

  • Vectors: Vectors are fundamental elements in linear algebra. They represent quantities that have both magnitude and direction and are typically denoted as column matrices. In a 2D space, a vector might be represented as (x, y), and in a 3D space, it would be (x, y, z).
  • Matrices: Matrices are rectangular arrays of numbers, symbols, or expressions. They are used to represent linear transformations and systems of linear equations. Matrices play a pivotal role in solving linear algebraic problems.
  • Vector Operations: Linear algebra defines operations on vectors, such as vector addition, scalar multiplication, and the dot product (inner product) of vectors. These operations are essential for manipulating and understanding vector quantities.
  • Matrix Operations: Linear algebra involves various matrix operations, such as matrix addition, matrix multiplication, and matrix inversion. These operations are critical for solving systems of linear equations and performing transformations.
  • Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are properties of square matrices that have numerous applications in linear algebra and various scientific fields. They help understand the behavior of linear transformations.

Calculus:

Calculus is a branch of mathematics that studies rate change and accumulation of quantities. It is a fundamental and essential field of mathematics with wide-ranging applications in science, engineering, economics, and many other disciplines. Here are the key concepts and components of calculus:

  • Derivatives: Calculus begins with derivatives, which represent the rate of change of a function at a given point. The derivative of a function f(x) at a point x is denoted as f'(x). It measures how the function’s value changes as the input (x) changes.
  • Limits: Limits are used to define derivatives precisely. They describe the behavior of a function as the input approaches a particular value. The limit of a function f(x) as x approaches a constant value ‘c’ is written as lim(x → c) f(x).
  • Integrals: Integrals are used to calculate the accumulation of quantities, such as area under curves or the total change in a function over an interval. The integral of a function f(x) is represented as ∫ f(x) dx.
  • Partial Derivatives: In multivariable calculus, derivatives are extended to functions with multiple variables. Partial derivatives describe how a function changes with respect to one variable while holding others constant.
  • Vectors and Vector Fields: Vector calculus deals with vectors and vector functions. It is essential in physics and engineering for studying phenomena with both magnitude and direction.

Numerical Methods:

Numerical methods are a set of mathematical techniques used to approximate solutions to mathematical problems that may not have exact analytical solutions. These methods are especially important in engineering, physics, computer science, and finance, where complex problems often require numerical approximations. Here are some key concepts and components of numerical methods:

  • Newton-Raphson Method: It is an iterative technique for finding the roots of real-valued functions. It involves approximating the function with a linear approximation and refining the estimate.
  • Lagrange Interpolation: Lagrange polynomials are used to approximate a function by constructing a polynomial that passes through given data points.
  • Simpson’s Rule: It provides a more accurate approximation of the integral by using quadratic approximations to the function.
  • Runge-Kutta Methods: These are higher-order methods for solving ODEs with improved accuracy and stability compared to Euler’s method.
  • Finite Difference Methods: These methods discretize PDEs by approximating derivatives with finite differences and solving the resulting system of equations numerically.
  • Finite Element Methods: FEM is used to approximate solutions to PDEs by dividing the problem domain into smaller elements and solving equations for each element.
  • Gradient Descent: It’s an iterative optimization algorithm used to find the minimum of a function, often used in machine learning and deep learning.
  • Monte Carlo Simulation: It is used for estimating complex probabilistic outcomes by simulating random events.

Probability and Statistics:

Probability and statistics are two closely related branches of mathematics that deal with uncertainty, randomness, and data analysis. They are essential tools in various fields, including science, engineering, social sciences, finance, and data science. Here’s an overview of probability and statistics:

 

  • Random Variables: Probability often deals with random variables, which are variables whose values depend on the outcome of a random event or process. For example, the outcome of rolling a fair six-sided die is a random variable.
  • Probability Distributions: Probability distributions describe how probabilities are assigned to various outcomes of a random variable. Common distributions include the normal distribution, binomial distribution, and Poisson distribution.
  • Conditional Probability: Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), representing the probability of event A given event B.
  • Bayesian Probability: Bayesian probability is a framework that updates probabilities based on new information or evidence. It is widely used in machine learning and artificial intelligence.
  • Sampling Methods: Sampling methods are crucial for selecting representative samples from a larger population. Simple random sampling, stratified sampling, and cluster sampling are common techniques.
  • Regression Analysis: Regression analysis examines the relationship between one or more independent variables and a dependent variable. Linear regression is a well-known technique for modeling these relationships.

Driving the change!

As mechanical engineers, we are uniquely positioned to bridge the gap between traditional engineering and AI. Our expertise in mathematics, honed through years of rigorous coursework, provides us with the tools to understand, appreciate, and contribute to the ever-evolving world of artificial intelligence. So, the next time you hear “AI,” remember that it’s not just a futuristic dream—it’s a mathematical reality. AI-driven design tools enable mechanical engineers to explore creative solutions for complex engineering challenges. In the future, AI and mechanical engineers will work hand in hand to develop autonomous vehicles and smart infrastructure. As mechanical engineers, we’re at the forefront of turning that dream into practical solutions for the world. Learn more about application of AI in mechanical engineer here.

 

You can also read similar article here to understand the importance of math in AI. 

Sania Shujaat

A Mechanical Engineer with a keen interest in applying AI to revolutionize Mechanical Engineering.

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