Lecturer	David Kay
Degrees	Schedule B1 (CS&P) — Computer Science and Philosophy Schedule A2 — Computer Science Schedule B1 — Computer Science Schedule A2(M&CS) — Mathematics and Computer Science Schedule B1(M&CS) — Mathematics and Computer Science
Term	Hilary Term 2025  (16 lectures)

Overview

Mathematical models have been used to investigate physical phenomena for centuries. Modelling is embedded in numerous scientific and non-scientific areas, including the fields of chemistry, physics, economics and social sciences, to name but a few. As the scientific world and ideas expanded, the systems have become more and more complex, moving from simple relationships between user inputs and observed outputs, to idealised differential equations and onto huge systems of highly non-linear partial differential equations. The major aim of this course is to present the concept of physics informed neural network approaches to approximate solutions systems of partial differential equations.

Learning outcomes

Fundamental mathematics of neural networks including approaches to physical models. Including minimising loss functions, partial differential of neural network input data and the building of partial differential operators, applications to mathematical models of the physical world and the critical evaluation of such methods.

Prerequisites

This course will require a firm knowledge of the material presented in the courses in
• Discrete Maths
• Linear Algebra
• Continuous Maths
• Machine Learning
• Competency in computer programming
Scientific Computing is non-essential, but will be complementary to this course.

Syllabus

Being able to comprehend and apply to basic models, the link between differential equations and the physical world. An understanding of classical approaches to numerical approximation of differential equations, in particular the difficulties that are encountered. Fundamentals of the mathematics of neural networks, including optimisation techniques and efficient calculations. The forming of a neural network to approximate the solutions of partial differential equations. This will include the issues of applying boundary and initial values set within the mathematical model of the physical system. The development of such networks to non-linear equations and multivalued outputs. Finally, applications and investigations within fluid flow and phase field models. 

Code development of PINN will be undertaken. This will be used to investigate the PINN approach to modelling PDEs. The choice of the code’s language to be used will be down to the individual, this will include any neural network packages you wish to use.

Reading list

• Linear Algebra and Learning from Data, G. Strang.
• Physics-Informed Machine Learning, G. E. Karniadakis, I. G. Kevrekidis, L. Lu, et
  al. in Nature Review Physics.
• Mathematics for Machine Learning by M. P. Deisenroth, A. A. Faisal and C. S, Ong.
• An introductory book on partial di↵erential equations is not essential as what is
  required for the course will be contained solely within the lecture notes. A good
  example is the book, Non-Linear Partial Di↵erential Equations for Scientists and
  Engineers, by L. Debnath.

Taking our courses

This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses

Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.