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Supriyo Datta and Hyunsoo Choi

In the physical world we are surrounded by waves of all kinds, each with its own underlying physics. Acoustic waves for example involve the vibrations of the medium which we can hear, while electromagnetic waves involve vibrations of electric fields that require no medium and are detected with specially designed antennas. These vibrations are physically very different and are usually taught in very different courses although they are governed by very similar partial differential equations. Furthermore, the equations are sufficiently opaque that they hide the underlying unity. Our aim is to make this unity apparent through a unifying viewpoint, using neural networks (NN) as a pictorial tool to visualize the underlying mathematics.

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In the physical world we are surrounded by waves of all kinds, each with its own underlying physics. Acoustic waves for example involve the vibrations of the medium which we can hear, while electromagnetic waves involve vibrations of electric fields that require no medium and are detected with specially designed antennas. These vibrations are physically very different and are usually taught in very different courses although they are governed by very similar partial differential equations. Furthermore, the equations are sufficiently opaque that they hide the underlying unity. Our aim is to make this unity apparent through a unifying viewpoint, using neural networks (NN) as a pictorial tool to visualize the underlying mathematics.

All waves can be viewed as a collection of individual oscillators, and so we will start in Week 1 by laying out the mathematical and conceptual framework for oscillatory systems as diverse as mechanical oscillators (mass-spring), electrical oscillators (inductor-capacitor), ferromagnets and quantum spins. The physical variables are very different and even the equations may look different, but we will stress their unifying characteristics, introducing the concept of normal modes and mapping them all to an NN model. In Week 2 , we will show how the coupling of different elementary oscillators leads to different wave systems, with quantitative illustrative examples.

This is a pilot 2-week course where we will illustrate the basic idea for acoustic, Schrödinger, and electromagnetic waves, all in Week 2. In future versions, our aim will be to expand the content with more illustrative examples as well as with more advanced topics such as quantized oscillators and waves.

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What's inside

Learning objectives

  • How diverse oscillatory systems can all be described by a common mathematical structure that can be visualized with a neural network-like pictorial representation.
  • The formation of wave systems by coupling individual oscillators along with the concepts of dispersion relation and group velocity.
  • How to perform quantitative calculations of the waves generated by sources.

Syllabus

Week 1: Oscillators
1.1. Math and the Physical World
1.2. Exponential Response
1.3. Oscillatory Response
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1.4. More Oscillator Examples
1.5. Impulse Response
1.6 Sinusoidal Excitation
Week 2: Waves
2.1. Oscillators to Waves
2.2. Normal Modes
2.3. Tx Lines and Guitars
2.4. Dipole Antenna

Good to know

Know what's good
, what to watch for
, and possible dealbreakers
Explores wave dynamics across multiple disciplines, uniting concepts from acoustics, electromagnetics, and quantum mechanics
Instructors Supriyo Datta and Hyunsoo Choi are recognized researchers in their respective fields of condensed matter physics and electrical engineering
Leverages neural networks to visualize the underlying mathematics, making complex wave phenomena more accessible
Suitable for learners with a strong foundation in mathematics, including differential equations
Focuses on the theoretical framework and mathematical modeling of waves, with limited practical applications

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Mathematics of Waves: Visualized with Neural Networks with these activities:
Review common mathematical functions and notation
This will strengthen skills necessary to understand the mathematical concepts within
Browse courses on Trigonometric Functions
Show steps
  • Review the definitions and graphs of basic functions
  • Practice manipulating algebraic expressions and equations
  • Solve problems involving trigonometry and calculus
Solve differential equations
Many of the underlying equations require a basic understanding of solving differential equations
Browse courses on Differential Equations
Show steps
  • Identify the type of differential equation
  • Find an appropriate solution method
  • Solve the differential equation
  • Interpret the solution
Attend an online workshop on wave propagation
Visual examples and discussion can help with understanding
Browse courses on Wave Propagation
Show steps
  • Find an online workshop on wave propagation
  • Register for the workshop
  • Attend the workshop and take notes
  • Review your notes after the workshop
Three other activities
Expand to see all activities and additional details
Show all six activities
Write a blog post about a specific type of wave
Can help to deepen the understanding of concepts and improve writing skills
Show steps
  • Choose a specific type of wave to write about
  • Research the topic and gather information
  • Organize your thoughts and write an outline
  • Write the blog post
  • Edit and publish the blog post
Mentor a student or colleague in understanding wave concepts
Explaining concepts to others reinforces understanding
Browse courses on Waves
Show steps
  • Identify a student or colleague who needs help understanding wave concepts
  • Set up a regular meeting time
  • Review the basics of wave concepts
  • Work through practice problems together
  • Provide feedback and support
Participate in a wave simulation competition
Practical application can solidify the knowledge of the underlying principles
Show steps
  • Find a wave simulation competition
  • Register for the competition
  • Develop a wave simulation model
  • Test and refine the model
  • Submit the model to the competition

Career center

Learners who complete Mathematics of Waves: Visualized with Neural Networks will develop knowledge and skills that may be useful to these careers:
Mathematical Engineer
Mathematical Engineers research, develop, and apply mathematical principles and techniques to solve problems in a variety of fields, including engineering, computer science, and finance. They use mathematical models to analyze and predict the behavior of complex systems. The topics covered in this course can help Mathematical Engineers build a strong foundation in the mathematics of waves, which is essential for understanding and solving problems in many different fields.
Data Scientist
Data Scientists use scientific methods, processes, algorithms, and systems to extract knowledge and insights from data in various forms, both structured and unstructured. They use neural networks to visualize and analyze data. The topics covered in the course can help Data Scientists build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Operations Research Analyst
Operations Research Analysts use advanced analytical techniques, such as mathematical modeling and optimization, to help organizations make better decisions. They use neural networks to visualize and analyze data. The topics covered in the course can help Operations Research Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Financial Analyst
Financial Analysts use mathematics to analyze and evaluate financial data. They use neural networks to visualize and analyze data. The topics covered in the course can help Financial Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing financial data.
Actuary
Actuaries use mathematics to assess and manage financial risks. They use neural networks to visualize and analyze data. The topics covered in the course can help Actuaries build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing financial risks.
Software Engineer
Software Engineers design, develop, and maintain computer software. They use neural networks to visualize and analyze data. The topics covered in the course can help Software Engineers build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Computer Scientist
Computer Scientists use mathematics to design, develop, and analyze software and hardware systems. They use neural networks to visualize and analyze data. The topics covered in the course can help Computer Scientists build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Quantitative Analyst
Quantitative Analysts use mathematics to analyze and evaluate financial data. They use neural networks to visualize and analyze data. The topics covered in the course can help Quantitative Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing financial data.
Risk Analyst
Risk Analysts use mathematics to assess and manage financial risks. They use neural networks to visualize and analyze data. The topics covered in the course can help Risk Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing financial risks.
Statistician
Statisticians use mathematics to collect, analyze, interpret, and present data. They use neural networks to visualize and analyze data. The topics covered in the course can help Statisticians build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Data Analyst
Data Analysts use mathematics to analyze and interpret data. They use neural networks to visualize and analyze data. The topics covered in the course can help Data Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Business Analyst
Business Analysts use mathematics to analyze and interpret data. They use neural networks to visualize and analyze data. The topics covered in the course can help Business Analysts build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Market Researcher
Market Researchers use mathematics to analyze and interpret data. They use neural networks to visualize and analyze data. The topics covered in the course can help Market Researchers build a strong foundation in the mathematics of waves, which is essential for understanding and analyzing data.
Economist
Economists use mathematics to analyze and interpret data. They use neural networks to visualize and analyze data. The topics covered in the course may be useful for Economists who want to build a strong foundation in the mathematics of waves, which can be helpful for understanding and analyzing economic data.
Financial Planner
Financial Planners use mathematics to analyze and interpret financial data. They use neural networks to visualize and analyze data. The topics covered in the course may be useful for Financial Planners who want to build a strong foundation in the mathematics of waves, which can be helpful for understanding and analyzing financial data.

Reading list

We've selected eight books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Mathematics of Waves: Visualized with Neural Networks.
An introductory textbook on waves and oscillations, covering both classical and quantum mechanics. provides a solid foundation for understanding the concepts of wave propagation and oscillatory systems, which are central to the course.
A comprehensive textbook on electromagnetism, providing a thorough understanding of the fundamental principles and applications of electromagnetic theory. is particularly valuable for its coverage of electromagnetic waves, which are a key topic in the course.
A classic textbook on optics, covering a wide range of topics from basic principles to advanced applications. offers a comprehensive and in-depth exploration of optical phenomena, which are closely related to electromagnetic waves.
A comprehensive textbook on mathematics for machine learning, providing a solid foundation for understanding the mathematical concepts underlying machine learning algorithms. is particularly valuable for its coverage of linear algebra and calculus, which are essential for understanding the course material.
A comprehensive textbook on numerical methods for partial differential equations, providing a solid foundation for understanding the numerical techniques used to solve wave equations. is particularly valuable for its coverage of finite difference and finite element methods.
A popular science book on quantum field theory, providing an accessible introduction to the subject for non-specialists. is particularly valuable for its clear explanations and engaging writing style.
A textbook on Maxwell's equations, providing a clear and concise introduction to the subject. is particularly valuable for its step-by-step explanations and practice problems.

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