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YZZ201 - Differential Equations (Autumn Term)

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Course Information

  • Venue: Class Z3, Department Floor, Annex Building, Faculty of Arts and Sciences

  • Date&Time: 09:15-12:00 on Thursdays

  • Objectives: This course aims to teach students differential equations (DEs) and their applications in artificial intelligence and machine learning along with several engineering disciplines.

  • Textbook: 

    • David V. Kalbaugh, Differential Equations for Engineers: The Essentials, 1st Ed., CRC Press, 2018.

  • Auxiliary Sources:

    • Joakim Sundnes, Solving Ordinary Differential Equations in Python, 1st Ed., Springer, 2024.​

    • Karline Soetaert, Jeff Cash, and Francesca Mazzia, Solving Differential Equations in R, 1st Ed., Springer 2012.

    • Clemens Heitzinger, Algorithms with JULIA: Optimisation, Machine Learning, and Differential Equations Using the JULIA Language, 1st Ed., Springer, 2022.

    • Christian Constanda, Differential Equations: A Primer for Scientists and Engineers, 2nd Ed., Springer, 2017.

    • Dennis G. Zill, Advanced Engineering Mathematics, 6th Ed., Jones & Bartlett Learning, 2018.

    • William E. Boyce, Richard C. Diprima, and Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 11th Ed., Wiley, 2017.

    • Brent J. Lewis, E. Nihan Onder, and Andrew A. Prudil, Advanced Mathematics for Engineering Students: The Essential Toolbox, 1st Ed., Elsevier, 2022.

    • Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud, Neural Ordinary Differential Equations, Advances in Neural Information Processing Systems (NeurIPS), 2018.

    • Christopher Rackauckas, SciML Book: Parallel Computing and Scientific Machine Learning, MIT (open access, online), 2020.

  • Contents: Introduction to DEs, First-Order Ordinary Differential Equations (ODEs), Second-Order Linear ODEs, Higher-Order Linear ODEs, The Laplace Transform, Systems of First-Order ODEs, Partial DEs and Series Solutions, Applications in AI&ML: Neural ODEs and Gradient Flow.

  • Learning Outcomes: Upon successful completion of this course, students will be able to:

    • Classify differential equations by order, linearity, and type (ordinary vs. partial), and identify the appropriate solution method for a given problem.

    • Formulate and solve first-order ODEs analytically using separation of variables, integrating factors, and exact equation techniques.

    • Solve second-order and higher-order linear ODEs using methods including undetermined coefficients, variation of parameters, and series solutions.

    • Apply the Laplace transform to solve linear ODEs with discontinuous or impulsive forcing functions, including initial value problems.

    • Formulate and solve systems of first-order ODEs, and analyse the stability and qualitative behaviour of their solutions.

    • Derive and interpret basic partial differential equations and their series-based solution methods, including separation of variables.

    • Implement numerical solvers for ODEs and systems of ODEs using at least one computational tool (Python, R, or Julia), and critically compare analytical and numerical solutions.

    • Explain the mathematical connection between differential equations and gradient-based optimisation methods used in machine learning, including the interpretation of gradient descent as a continuous-time dynamical system.

    • Describe the formulation of Neural ODEs and articulate their relevance to continuous-depth deep learning architectures.

    • Evaluate the applicability of differential equation-based modelling to engineering and artificial intelligence problems, and select suitable analytical or computational approaches for novel problem settings.

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Lecture Notes​

  1. Course Introduction and Scope, Foundations of DEs

  2. First-Order Linear ODEs

  3. First-Order Nonlinear ODEs

  4. Existence and Uniqueness

  5. Second-Order Linear ODEs

  6. Higher-Order Linear ODEs

  7. Higher-Order Linear ODEs

  8. Midterm Examination

  9. Laplace Transforms

  10. Systems of First-Order ODEs

  11. Systems of First-Order ODEs

  12. Partial DEs and Series Solutions

  13. Partial DEs and Series Solutions

  14. Applications in AI&ML: Neural ODEs and Gradient Flow

  15. Applications in AI&ML: Neural ODEs and Gradient Flow

  16. Final Examination​

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Exams

  • Week 9: Midterm Examination (2026) (2022) (2021)

  • Week 16: Final Examination (2027) (2023) (2022)

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Announcements

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