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Mathematical Techniques & Machine Learning - PHY00059I

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  • Department: Physics
  • Credit value: 20 credits
  • Credit level: I
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

Mathematical Techniques

Numerical and analytic solutions to partial differential equations is the bedrock of modern theoretical and computational physics. The aim of this course is to learn how to solve complex mathematical and physical problems both analytically and computationally. This module will introduce how a computer can be used to solve differential equations which describe fundamental physical processes, such as heat flow or the evolution of the wavefunction of quantum mechanics. You will learn powerful mathematical techniques, such as the ‘calculus of variations’ and ‘complex analysis’ – elegant and vital tools for theoretical physics and beyond.

Machine Learning

This component introduces the basics of machine learning, with a focus on regression and optimization. As such, the mathematical concepts involved will be calculus, statistics and linear algebra.

Related modules

Pre-requisites:  Mathematical, Computational and Professional Skills I and II or equivalent

Module will run

Occurrence Teaching period
A Semester 1 2024-25

Module aims

Numerical and analytic solutions to partial differential equations is the bedrock of modern theoretical and computational physics. The aim of this course is to learn how to solve complex mathematical and physical problems both analytically and computationally. This module will introduce how a computer can be used to solve differential equations which describe fundamental physical processes, such as heat flow or the evolution of the wavefunction of quantum mechanics. You will learn powerful mathematical techniques, such as the ‘calculus of variations’ and ‘complex analysis’ – elegant and vital tools for theoretical physics and beyond.

In this introduction to machine learning, you will cover topics such as linear regression, decision trees and neural networks. As such, the mathematical concepts involved will be calculus, statistics and linear algebra. Theoretical ideas will be illustrated by real-world examples from different areas of physics. Practical sessions will be taught using various Python-based modern machine learning libraries (e.g. TensorFlow and/or PyTorch).

Module learning outcomes

  • Apply finite difference methods to solve advanced differential equations with the computer and classify these solutions

  • Apply complex analysis to determine whether a function is analytic and evaluate advanced integrals

  • Find the minima and maxima of a multivariable function subject to constraints

  • Recognise ‘special functions’, such as Bessel and Legendre functions, and use these functions to form a complete set

  • Choose, implement, and evaluate machine learning algorithms

  • Train machine learning algorithms and evaluate the performance of those models

  • Use a machine learning library and apply to real datasets

Module content

Mathematical Techniques

  • Finite difference representation of first, second order derivatives: apply to physical processes such as diffusion; use simple explicit methods, e.g. Euler; determine stability, convergence criteria and consistency; explicit and implicit methods for the wave equation

  • Complex Variable Techniques: Cauchy-Riemann relations and Cauchy's theorem; Laurent expansion; Cauchy residue theorem; Contour integrals; Physical examples of contour integration.

  • Variational Techniques, Differential Equations, and Special Functions: Lagrange multipliers and Euler-Lagrange equations; Bessel's equation and the properties of Bessel functions of the first and second kind; Legendre's equation and Legendre polynomials; the definition of a cmplete set with examples.

Machine Learning

  • Overview of ML methods - ML vs AI, supervised vs unsupervised learning, classification vs regression

  • Data preparation and big data

  • Linear regression, logistic regression and multivariate optimization

  • Overfitting and regularisation

  • Neural networks and back propagation

  • k-means clustering

  • Decision trees and non-parametric classification

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 40
Essay/coursework 40
Essay/coursework 10
Essay/coursework 10

Special assessment rules

Other

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 40
Essay/coursework 40

Module feedback

'Feedback’ at a university level can be understood as any part of the learning process which is designed to guide your progress through your degree programme. We aim to help you reflect on your own learning and help you feel more clear about your progress through clarifying what is expected of you in both formative and summative assessments.

A comprehensive guide to feedback and to forms of feedback is available in the Guide to Assessment Standards, Marking and Feedback. This can be found at:

https://www.york.ac.uk/students/studying/assessment-and-examination/guide-to-assessment/

The School of Physics, Engineering & Technology aims to provide some form of feedback on all formative and summative assessments that are carried out during the degree programme. In general, feedback on any written work/assignments undertaken will be sufficient so as to indicate the nature of the changes needed in order to improve the work. Students are provided with their examination results within 25 working days of the end of any given examination period. The School will also endeavour to return all coursework feedback within 25 working days of the submission deadline. The School would normally expect to adhere to the times given, however, it is possible that exceptional circumstances may delay feedback. The School will endeavour to keep such delays to a minimum. Please note that any marks released are subject to ratification by the Board of Examiners and Senate. Meetings at the start/end of each semester provide you with an opportunity to discuss and reflect with your supervisor on your overall performance to date.

Our policy on how you receive feedback for formative and summative purposes is contained in our Physics at York Taught Student Handbook.

Indicative reading

Mathematical Techniques

  1. Complex Analysis with Applications Book by Loukas Grafakos and Nakhle H. Asmar

  2. Introduction to the Calculus of Variations Book by Bernard Dacorogna

  3. Special Functions of Mathematical Physics: A Unified Introduction with Applications Book by NIKIFOROV and Vasili B. Uvarov

Machine Learning

Machine Learning by Tom Mitchell, McGraw-Hill


The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.