- Department: Physics
- Module co-ordinator: Dr. Emma Osborne
- Credit value: 20 credits
- Credit level: I
- Academic year of delivery: 2023-24
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.
Pre-requisites: Mathematical, Computational and Professional Skills I and II or equivalent
Occurrence | Teaching cycle |
---|---|
A | Semester 1 2023-24 |
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).
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
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
Task | Length | % of module mark |
---|---|---|
Closed/in-person Exam (Centrally scheduled) Mathematical Techniques & Machine Learning |
2 hours | 40 |
Essay/coursework Machine Learning Assignment |
N/A | 40 |
Essay/coursework Machine Learning Mini Assignment |
N/A | 10 |
Essay/coursework Mathematical Techniques Practice Questions |
N/A | 10 |
Non-reassessable
Task | Length | % of module mark |
---|---|---|
Closed/in-person Exam (Centrally scheduled) Mathematical Techniques & Machine Learning |
2 hours | 40 |
Essay/coursework Machine Learning Assignment |
N/A | 40 |
'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.
Mathematical Techniques
Complex Analysis with Applications Book by Loukas Grafakos and Nakhle H. Asmar
Introduction to the Calculus of Variations Book by Bernard Dacorogna
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