Undergraduate Teaching 2025-26

2025-26

2025-26

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Engineering Tripos Part IIB, 4C11: Data-driven and learning based methods in mechanics and materials, 2025-26

Leader

Dr B Liu

Lecturer

Dr B Liu and Dr A Cicirello

Timing and Structure

Lent term. 13 lectures. Assessment 100% coursework

Prerequisites

3C7 and 3D7 useful

Aims

The aims of the course are to:

  • Introduce the state-of-the-art concepts and theories for deep learning and neural networks.
  • Describe the main methods of constructing learning-based partial differential equation solvers with illustrative examples on Darcy flow and elasticity.
  • Explain the concept and theory of path dependency (memory) and multi-scale modelling, with application of the data-driven methods for discovering and approximating constitutive models for various materials.

Objectives

As specific objectives, by the end of the course students should be able to:

  • Understand the principles of applying data-driven methods to physical problems.
  • Design, implement and train learning-based PDE solvers for stress analysis.
  • Discover non-linear, path-dependent material models from data using deep neural networks.

Content

Mechanics and materials are gradually becoming data-rich due to rapid advances in experimental science and high-performance multiscale computing. There has been a growing interest in the field of solid mechanics for developing data-driven and learning-based methods to characterize, understand, model, and design material/structural systems. With data-driven approaches, it is possible to remove/relax the need for ad hoc constitutive models for describing the material behavior, to achieve fast multi-scale computation for structures as well as to generate optimal designs. This module will introduce a wide spectrum of data-driven/learning based methods that have been developed and used in mechanics and materials, with an emphasis on developing a working understanding of how to apply these methods in practice.

Syllabus

Neural network basics (4L)

  1. Basic concepts in supervised and unsupervised learning.
  2. Fully connected neural network, stochastic gradient descent.
  3. Advanced neural network architectures: convolution neural network, Res-net, U-net.
  4. Python for machine learning and pytorch tutorial.

 

Machine learning for PDEs: Physics Informed Neural Networks and Neural Operators (4L)

  1. Physics informed neural networks for ODE and PDE.
  2. Learning the solution operator of PDE with Neural Operators.
  3. Fourier and Graph Neural Operators.

 

Machine learning for path dependent problems and learning based multi scale modeling (4L))

  1. Machine learning methods for memory and path dependence.
  2. Long Short Term Memory and Transformer networks.
  3. Multiscale modeling and Recurrent Neural Operator.
  4. Generative modeling methods.

 

Data-driven methods in mechanics and beyond - guest lecture (1L)

  1. Neural operators in climate change - the earth 2 project.
  2. Researches in NVDIA.

 

Coursework

Course work 1: Neural network and Pytorch basics

Description: This course work consists of two problems: 

(i) Regression problem: Student will be provided with measured stress-strain data for two unknown elastic materials. Students are asked to build, train and validate a neural network model for approximating the constituitive relationship of the material. They will use basic fully connected neural network. 

(ii) Classification problem: Student will be asked to design, implement and train a neural network classifier that predicts whether a truss structure (Effiel tower) will collapse under certain external pressures. They will investigate the use of both basic fully connected neural network as well as advanced deep Res-net, and assess the netowrks performance.

Format: 1 individual report

Course work 2: Learning based stress analysis

Students will be asked to solve a 2D elasticity problem for a plate with hole under bi-axial loading using Physics Informed Neural Networks. They will also be asked to design and implement a Fourier Neural Operator to learn the solution operator of the Darcy flow problem. 

Format: 1 individual report

Course work 3: Learning based constitutive model for anisotropic solids

Description:  Students will be asked to come up with novel designs of neural network architectures that can represent memory/path-dependency of solid materials. They will be given a micro-mechanical unit-cell problem governed by visco-elasticity, and are expected to train their neural networks to find the homogenized macroscopic constitutive model, together with the hidden internal variables that captures the memory of deformation path at the macroscopic scale. 

Format: 1 individual report

 

Booklists

Please refer to the Booklist for Part IIB Courses for reference to this module, this can be found on the associated Moodle course.

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 04/06/2025 13:26

Engineering Tripos Part IIA Project, SG2: Bioreactor Control, 2025-26

Leader

Dr S Bakshi

Timing and Structure

Fridays 11-1pm and Tuesdays 9-11am plus afternoons

Prerequisites

2P6, 3F1 (desirable), 3G1 (desirable)

Aims

The aims of the course are to:

  • To gain understanding of the relevant biological processes and process control in bioreactors
  • To learn about the operation and calibration of the relevant sensors and actuators for monitoring and maintaining process variables
  • To design an experiment to analyse the role of process variables on system performance

Objectives

As specific objectives, by the end of the course students should be able to:

  • To develop a virtual bioreactor model for simulating different controllers and associated parameters
  • To use and calibrate sensors for cell density and temperature of the cell culture in a microbial bioreactor
  • To regulate one environmental variables (e.g. temperature) and cell density for optimising growth of the culture
  • To model and experimentally test microbial population growth under nutrient limited conditions at controlled temperature
  • To implement and compare performance of open-loop and closed-loop control of cell density to regulate nutrient availability

Content

 
BACKGROUND:
 
Bioreactors are the key technology for bioprocess engineering. Primarily, bioreactors are used to keep cells (microbial or mammalian) under controlled conditions such that they can optimally perform the desired tasks. Example application include bioproduction of antibodies and vaccines, tissue engineering, or even nutrient production usign bacteria and algae.
 
PROJECT: 
 
This project introduces you to some of the essential concepts of the bioprocesses in microbial bioreactors and how to use sensors and actuators for monitoring and controlling the environmental variables to keep those bioprocesses operating in an efficient manner. You will also learn about sources of noise and drift in such bioprocesses and how closed-loop feedback control can be implemented for maintaining the process variables. You will develop a virtual bioractor which incorporates the relevant processes (preferably in MATLAB) and can enable testing control performance. You will use experimental data to test the model predictions. 
 
The project covers concepts of logistic growth of microbial populations, scattering based measurements of population growth over time and single cell imaging for calibration of such measurements, and how temperature, nutrient density, and oxygen level affect population growth. For process control, the project will cover chemostat and turbidostat modes of culture maintenance.  
 
FORMAT:
 
Students will work in pairs. Engineers might be paired with medics. There are total 4 lab sessions. Each student will write interim reports by the end of weeks 1, 2, and 3 and a final report by the end of week 4.
 
ACTIVITIES: 
 
Week 1: Develop and test a temperature regulation simulator for the bioreactor 
Week 2: Monitor and model cell growth at regulated temperature
Week 3: Test different cell density regulation strategies at regulated temperature and explain the observed performance differences
Week 4: Develop an integrated simulator for the bioreactor cell density regulation

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 01/12/2025 07:27

Engineering Tripos Part IIA Project, GG3: Neural data analysis, 2025-26

Leader

Prof G Hennequin

Timing and Structure

Students work to their own schedule. A staffed "surgery" runs every weekday to give help, advice and feedback.

Prerequisites

Part I computing; Either of 3F3 or 3F8

Aims

The aims of the course are to:

  • To introduce students to machine learning approaches to modeling of natural phenomena and hypothesis testing.
  • To apply generative modeling techniques to neurobiological data in order to infer underlying mechanisms.
  • To gain practical experience with model inference and validation, and hypothesis testing via model selection.
  • To gain experience with issues such as overfitting, and possible lack of robustness of conclusions due to model misspecification.

Content

In this projects students will study two proposed mechanisms hypothesised to underlie the firing rate patterns of neurons recorded in an area of the monkey cortex thought to be involved in evidence integration for decision making. The ultimate goal is to infer which of the two mechanisms (as two competing hypotheses) had generated a simulated dataset of neural responses provided to the students.

The students will also be provided with a package written in Python allowing them to simulate two generative models as concrete formalisations of the two conceptual hypotheses.
The students will first explore the behaviour of the outputs (neural spike trains) of these two models in different regions of their parameter space. They will be guided to appreciate
how some key aspects of the data, commonly relied on in neuroscience, can look near identical in the data generated by the two simulators despite their qualitatively different mechanisms.
This motivates the use of Bayesian statistical techniques for inferring the models and their parameters from whole datasets (as opposed to summary statistics).

Students will then carry out model fitting and Bayesian inference of latent variables and model parameters. This is partly done by writing their own code, and partly using provided Python programmes.
Students will also carry out model validation using simulated data generated by ground-truth models, in order to gain insight into factors affecting model recovery and overfitting, and approaches for mitigating it.
Students will explore the issue of "brittleness" and non-robustness of hypothesis testing, when auxiliary features of the models formalising the hypotheses do not match those in the ground-truth model.  Students will then apply their gained knowledge to infer the mechanism underlying a dataset of neural responses.

Format

Week 1

Explore the behaviour of the two generative models and the effect of different parameters. Understand how and why trial-average firing rates generated by the two models can look similar.

Week 2

Students will generate datasets and carry out model inference (inference of parameters) by implementing the expectation-maximization and variational inference algorithms. Students will assess overfitting using cross-validation and study its behaviour with growing dataset size.

Week 3

Students will be introduced to information criteria for model selection, and will carry out model-recovery experiments to assess whether a given dataset allows for reliable inference of underlying model.

Week 4

Students will explore simulating and fitting models which realise the same conceptual mechanisms but differ in other aspects, in order to explore the effect of those differences and mismatches on model selection. At the end students apply their gain experience to infer the mechanism that generated a dataset provided to them.

Coursework

Coursework Due date Marks
Interim report Beginning of 2nd week 20
Final report Friday of 4th week 60

 

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 12/05/2026 09:56

Engineering Tripos Part IIB, 4G7: Control and computation in living systems, 2025-26

Leader

Timothy O'Leary

Second Assessor

Fulvio Forni

Timing and Structure

Michaelmas term, 12 Lectures + problems classes. 100% Exam

Prerequisites

Ability to program numerical simulations in MATLAB or Python. No formal prerequisites but 3G2 Mathematical Physiology and 3G3 Intro to Neuroscience would be very useful.

Aims

The aims of the course are to:

  • Introduce students to formalisms for modelling biological systems at multiple levels, from molecules to organisms
  • Provide tools for understanding how nonlinear computations arise in biological systems to enable decision making, timing, memory and control
  • Develop an appreciation of current research in quantitative biology through case studies of recent and/or classic research papers

Objectives

As specific objectives, by the end of the course students should be able to:

  • Introduce examples of biological computation and control: bacterial chemotaxis, circadian oscillators, motor pattern generators, biochemical
  • Construct and analyse formal models of living systems, including biochemical networks, neural networks and populations of agents
  • Provide a contextual introduction to key mathematical and computational tools: (nonlinear) feedback control, qualitative theory of ODEs, singular perturbation theory, stochastic dynamical systems, simulation methods.
  • Develop ability to simulate and experiment with models of living systems and report results coherently and critically
  • Develop ability to read, understand and appreciate/contextualise research papers in quantitative biology and mathematical biology

Content

Living systems, including single cells, nervous systems and animal/human populations, are increasingly well understood in terms of the computations they perform and the control principles they embody. This has enabled a paradigm shift in bioengineering, allowing us to pick apart and understand how living systems function and, crucially, manipulate and exploit these functions in a principled way.

 

This course will introduce students to current research in this field and provide tools and examples for analysing, modelling and designing biological and biologically-inspired systems. It therefore fills an important component of an up to date bioengineering curriculum and complements several courses on offer in Bioengineering (4G1 Mathematical Biology of the Cell, 4G3 Computational Neuroscience) and Information Engineering (4F2 Nonlinear and Robust Control, 4M7 Practical Optimization). It will naturally complement projects and modules in bioengineering and neuroscience.

Course content (individual lectures may vary)

  1. Introduction to modelling formalisms with examples (mass action kinetics, agent/population dynamics, timescale separation)
  2. Switches and hysteresis: the fundamental motif for decision making and memory
  3. Introduction to phase plane analysis and qualitative theory of ODEs
  4. Gradient following algorithms in nature, chemotaxis
  5. From switches to pulses and nonlinear oscillations: the Fitzhugh Nagumo reduction of action potentials
  6. Consensus and decision making in populations of cells and animals
  7. Selected topics in biological control and computation and bio-inspired computation (e.g. brain machine interfaces, synthetic biochemical circuits, neuromorphic computing)

Coursework

Optional (unassessed) coding exercises, assigned reading.

Booklists

The following textbooks are useful

Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC press.

Berg, H. C. (2008). E. coli in Motion. Springer Science & Business Media.

Alon, U. (2006). An introduction to systems biology: design principles of biological circuits. Chapman and Hall/CRC.

 

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 17/06/2025 14:38

Engineering Tripos Part IIB, 4A15: Acoustics, 2025-26

Module Leader

Dr A Agarwal

Lecturers

Dr A. Agarwal and Dr W. Graham

Timing and Structure

Lent term: 16 lectures + 2 examples classes; Assessment: 100% exam

Prerequisites

No prerequisites. The module would be of interest to students with Aero, Mechnical, Bio or Civil Engineering background.

Aims

The aims of the course are to:

  • analyse and solve a range of practical engineering problems associated with acoustics.

Objectives

As specific objectives, by the end of the course students should be able to:

  • understand what sound is and how we perceive it
  • understand how sound is generated and propagated
  • understand the acoustics of a wide range of music and noise production

Content

We will analyse and solve a range of practical engineering problems associated with acoustics. Examples include modelling of noise sources from jets, fans, musical instruments, human voice, kettles, dripping taps, whistling mice, singing flames, etc. We will also study ways to reduce noise either at the source or through acoustic damping. Upon completion of this module, the students would be well placed to pursue academic research in the area of acoustics and related fields or to work in industry (the topics covered in the course is of interest to GE, Rolls-Royce, Airbus, Dyson, Mitsubishi Heavy Industries, automotive companies, music and biomedical industries, and acoustic consultancies).

 

What is sound and how does it propagate? (5L) (Dr A Agarwal)

  • Introduction
  • The wave equation
  • Some simple 3D wave fields (plane waves, surface waves and spherical waves)
  • Sound transmission through different media

Simples sounds sources (2L) (Dr A Agarwal)

  • Pulsating sphere
  • Oscillating sphere
  • Example: loudspeaker with and without a cabinet

General solution to wave eqn (2L) (Dr. A Agarwal)

  • Green's function
  • Sound from general mass and force sources (examples, Bliz siren and singing telephone wires)
 

Jet noise (Dr A Agarwal) (1 L)

  • Scaling of jet noise. How much does jet noise increase by if we double the jet's velocity?
  • What do jets and tuning forks have in common?
  • Lighthill's acoustic analogy
  • Sound of aircraft jets and handdriers 

Duct acoustics (2 L) (Dr A Agarwal)

  • Rectangular ducts (example, sound box)
  • Low-frequency sound in ducts
  • Circular ducts
  • Acoustic liners (Helmholtz resonator, blowing over a beer bottle)
 

Musical acoustics & everyday things (3L) (Drs A Agarwal)

  • String instruments 
  • Wind instruments 
  • Brass instruments 
  • Whistling of steam kettles and Rayleigh's Bird Call
  • Acoustics of dripping taps
 
 

Vocalisation (0.5 L) (Dr A Agarwal)

  • Human speech, singing and overtone singing
  • Mice mating calls
 

Fan noise (1L) (Dr A Agarwal)

  • Rotor alone noise
  • Rotor-stator interaction noise
 

Thermoacoustics instability (0.5 L) (Dr A Agarwal)

  • Rijke tube experiment (singing flames)

Booklists

Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.

Examination Guidelines

Please refer to Form & conduct of the examinations.

UK-SPEC

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

GT1

Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.

IA1

Apply appropriate quantitative science and engineering tools to the analysis of problems.

IA2

Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.

KU1

Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.

KU2

Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.

E1

Ability to use fundamental knowledge to investigate new and emerging technologies.

E2

Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.

E3

Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.

P1

A thorough understanding of current practice and its limitations and some appreciation of likely new developments.

P3

Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).

US1

A comprehensive understanding of the scientific principles of own specialisation and related disciplines.

US2

A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.

US4

An awareness of developing technologies related to own specialisation.

 
Last modified: 04/06/2025 13:24

Engineering Tripos Part IIB, 4M26: Algorithms and Data Structures, 2025-26

Module Leader

Prof Per Ola Kristensson

Lecturers

Prof Per Ola Kristensson, Dr A Tewari, Dr E Wu

Timing and Structure

Lent term. 16 lectures (including two integrated examples classes). Assessment: 100% exam.

Aims

The aims of the course are to:

  • Introduce the principles behind algorithm and data structure design and evaluation.
  • Cover key topics including elementary and advanced data structures, including sorting algorithms, graph algorithms, and so on.
  • Provide an understanding of how to translate algorithms into code for selected engineering problems through coding-focused computerised examples papers.

Objectives

As specific objectives, by the end of the course students should be able to:

  • Analyse the computational efficiency of algorithms.
  • Re-implement and debug algorithms.
  • Correctly choose a suitable algorithmic solution and set of data structures for a computational problem.
  • Understand the theoretical and practical advantages and disadvantages of various algorithmic approaches and established solutions.
  • Devise and implement new algorithms and data structures, or modify existing algorithms and data structures, to solve previously unencountered tasks.

Content

  • Part 1: Fundamentals of Algorithms and Data Structures (7L + 1 Example Class)
    • Interpreting and writing pseudocode, demonstrating correctness, arriving at tight/lower/upper bounds of running time/storage, and solving computational problems using a repertoire of data structures and algorithmic approaches.
  • Part 2: Algorithms and Data Structures in Engineering (7L + 1 Example Class)
    • Translating pseudocode into code, debug implementations of algorithms and data structures, apply algorithms and data structures to solve a range of frequent engineering problems, such as finding shortest paths, resource allocation, and scheduling.

Booklists

Introduction to Algorithms (3rd ed) by Cormen, T., Leiserson, C., Rivest, R., Stein, C. The MIT Press. ISBN:978-0-262-03384-8.

Also, please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 04/06/2025 13:33

Engineering Tripos Part IIB, 4G9: Biomedical Engineering, 2025-26

Module Leader

Dr T Bashford

Lecturers

Prof M Sutcliffe (MPFS), Dr T Bashford (TB), Prof T Makin (TM), Prof A Flewitt (AJF)

Timing and Structure

11 lectures; four discussion meetings. Assessment: 100% coursework. Lectures will be recorded.

Aims

The aims of the course are to:

  • Provide a comprehensive overview of biomedical engineering
  • Outline the key principles of good engineering design in a biomedical context
  • Introduce the concept of system design approach for sustainable improvement
  • Describe the technology adoption pathway in healthcare

Objectives

As specific objectives, by the end of the course students should be able to:

  • Conduct research and define the issues with existing medical devices or clinical procedures
  • Understand how to apply engineering knowledge to solve biomedical challenges
  • Communicate and work with healthcare professionals to validate the engineering designs
  • Use a broader systems design toolkit to address larger and more complex issues in healthcare

Content

The course has four case studies. Students will 'major' on one case study, but will need to attend (either in person or via recorded lectures) the lectures pertaining to the other case studies to cover all the required elements of the course.

General introduction (3L total) [TB (2L); MPFS (0.33L); GMB (0.33L); AJF (0.33L)]

Introduction of biomedical engineering and systems approach to systems improvement; introduction of four case studies

Monitoring after brain injury case study (2L) [TB]

Monitoring after brain injury; novel technology; stakeholder acceptance regulatory pathway.

Biomechanics case study (2L) [MPFS]

Knee biomechanics/kinematics; design for the knee replacement; clinical/patient acceptance

Wearable motor augmentation case study (2L) [TM]

Neurological, neuroanatomical and user considerations in the design of augmentation technology,  Basics of anatomy, user needs, patient and public engagement, and rapid iterative design cycling.

Biosensor case study (2L) [AJF]

Concept of point-of-care; microfluidic platform-assisted biosensors; manufacturing

Discussion meetings (5L) [Guest mentors (2L); all lecturers (3L)]

Short presentation sessions from guest mentors (University, NHS, industry) and panel discussions; open discussion meetings with lecturers

Coursework

Coursework Format

Due date

& marks

Initial coursework mapping 'canvas'

One-page document focusing on the big picture of the chosen case study

Learning objective:

  • demonstrate the framework of systematic engineering design
  • encourage the student to plan the case study by raising questions
  • adapt a genetic system design framework to a specific project at a high level
  • make an initial list of foci under each key topic on the canvas template

Individual Report

anonymously marked

End of week 2

[10%]

Expanded courswork mapping ' canvas'

A much expanded version of the first coursework element

Learning objective:

  • provide further guidance on the canvas on the activities that need to be considered by providing example questions
  • reflect on an accurate problem identification, risk management, the interdependency between technical and social components in the project

Individual Report

anonymously marked

End of week 5

[30%]

Final report

Final report - 20 page upper limit, 3,000 word upper limit

Learning objective:

  • provide information on the problem formulation, requirement specification, design, risk assessment, stakeholder acceptance, marketing/policy strategy, design solution, etc.

Individual Report

anonymously marked

End of week 9

[60%]

 

Booklists

Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 16/03/2026 17:09

Engineering Tripos Part IIB, 4D2: Advanced Structural Design, 2025-26

Module Leader

Prof. Simon Guest

Lecturers

Prof S Guest and Prof W Baker

Timing and Structure

Lent term. 16 timetabled sessions (lectures + design sessions). Assessment: 100% coursework.

Prerequisites

3D3 and 3D4 assumed

Aims

The aims of the course are to:

  • Instil an intuitive approach to structural design.
  • Introduce advanced concepts related to the design of structures.

Objectives

As specific objectives, by the end of the course students should be able to:

  • Design a wide variety of structures which meet both aesthetic and efficiency criteria.
  • Describe the relationship between form and force.
  • Manipulate structural geometry or forces to improve the structural behaviour.
  • Describe Airy stress functions and be able to design structures using them.
  • Describe the relationship between states of self-stress and mechanisms.
  • Design prestressable structures which contain mechanisms.
  • Describe the load path and how to optimise it.
  • Use Lagrange multipliers in constrained optimisation problems in structural design.
  • Understand the requirements for minimal total structural volume.
  • Describe stiffness and stability from a geometrical perspective.
  • Design gridshells and nets. Understand the unique behaviour of each.
  • Intuitively understand structural behaviour so that visual design can occur.
  • Describe the implications a structure’s design has on the stakeholders.

Content

Content and delivery will be largely provided by Prof Bill Baker. Prof Baker is the consulting partner at Skidmore Owings and Merrill in Chicago and Honorary Professor of this department. He is the world's leading structural engineer for the design of buildings and has been responsible for the design of many of the world's more iconic buildings. Prof Baker will teach the skills needed to become a proficient structural designer. The course aims to inform students about powerful new design tools which are growing in popularity throughout industry, many of which have been developed by Prof Baker in collaboration with this department.

Introduction to the course

A short history of structures and architecture. The importance of geometry and design. Discussion of its wide-reaching impacts and implications.

Graphic statics

Relationship between the form and force. How to design structures so that the forces flow where the designer wants them to.

Maxwell load path theorem

What it is and how it relates to the total volume of structural material used.

Force density

How we can use it to solve linear problems to find the geometry.

Virtual work and energy sizing

How it may be used to optimise the structural geometry, using Lagrange multipliers.

Michell trusses

Optimal structures and their behaviour.

Form finding for trusses

Discussion of the tools available to optimise topology, shape and size for structure.

Mechanisms and states of self-stress

The geometric relationship between mechanisms and states of self-stress, and the Maxwell-Calladine and Euler counts to obtain structurally sound trusses.

Geometric stiffness

The stiffness of structures and mechanisms is considered using force density. A short introduction to rigidity theory.

Airy stress functions and their application to truss design

To identify states of self-stress and mechanisms. The use of funiculars to include external loading.

Design of Gridshells and Cable Nets

How to design shells and gridshells using the Airy stress function and force density. The importance of obtaining planar faces and torsion free nodes. We also consider the design of tension structures, using prestress to stabilise mechanisms.

Coursework

Coursework Format

Due date

& marks

Coursework 1: (CW1) Fundamental theory

Answers to be submitted to a set of open-ended questions on theoretical topics.

Individually submitted

Anonymously marked

Due dates:
4pm Wed 4th Feb 2026
4pm Wed 11th Feb 2026

(on Moodle) 

[20/60]

Coursework 2: (CW2) Bridge Design

Students will design a discrete bridge structure using the tools developed in the course, choosing from a list of possible scenarios.

Individual report

Anonymously marked

Due date:
 4pm Wed 4th Mar 2026

(on Moodle)

[20/60]

Coursework 3: (CW3 ) Roof Design

Students will design an innovative roof structure using the tools developed in the course; they will decide on the structural system of the roof and choose from a list of possible venues.

Individual report

Anonymously marked

 Due date:
4pm Wed 1st Apr 2026

(on Moodle)

[20/60]

 

Booklists

Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 14/01/2026 07:04

Engineering Tripos Part IIB, 4M24: Computational Statistics and Machine Learning, 2025-26

Leader

Prof M Girolami

Timing and Structure

Michaelmas term. 75% exam / 25% coursework. Lectures will be recorded.

Prerequisites

3F3, 3F8, 3M1

Aims

The aims of the course are to:

  • Introduce students to foundational theoretical concepts and methodological tools essential for the successful development, analysis, and application of advanced Machine Learning and Computational Statistical methods.

Objectives

As specific objectives, by the end of the course students should be able to:

  • Introduce the students to the required statistical and mathematical concepts that underpin all rigorously designed Machine Learning and Computational Statistical methods that can be used practically across all the contemporary engineering sciences
  • Introduce the students to advanced computational statistical inference methods required to design Machine Learning solutions to a range of challenging large scale engineering problems where data and models are synthesised

Content

By successful completion of this course the student will have an appreciation and basic understanding of the mathematical, probabilistic, and statistical foundations of modern Computational Statistical Methods and recent developments in Machine Learning algorithms.

Computational Statistics and Machine Learning

Lecture.1. Monte Carlo Methods - A : Numerically computing integrals, the law of large numbers for Monte Carlo estimators, The Central Limit Theorem for Monte Carlo estimators.

Lecture.2. Monte Carlo Methods - B : Improving MC estimators, Importance Sampling, Control Variates to reduce variance of estimates.

Lecture.3. Lebesgue Integral and Measure - A : Difference between Riemann and Lebesgue Integral, why Lebesgue integral is required for machine learning and engineering, definition of Lebesgue integral.

Lecture.4. Lebesgue Integral and Measure - B : Definition of Lebesgue Measure, Radon-Nikodym derivative and change of measure, Measure theoretic basis of Probability (Kolmogorov), Random Variables.

Lecture.5. Markov Chain Monte Carlo - A : Definition of Markov chain and invariant distributions, presentation of the Metropolis and Hastings method.

Lecture.6. Markov Chain Monte Carlo - B : Metropolis Hastings in multiple dimensions, the Gibbs Sampler.

Lecture.7. Vector, Metric, and Banach Spaces : generalisation of Euclidean space in R^3 to infinite dimensional spaces, Completion of space and definition of Banach space of functions.

Lecture.8. Hilbert Spaces : Inner product space, definition of Hilbert space, Cauchy sequences and function approximation, Reproducing kernel Hilbert Space and function approximation.

Lecture.9. Sobolev Spaces : Definition of weak derivatives, understanding rates of convergence of function approximations based on properties of Sobolev space (smoothness)

Lecture.10. Gaussian Measure in Hilbert Space : Illustrating non-existence of Lebesgue Measure in function space, construction of finite Gaussian measure in Hilbert space, definition of Bayes rule (via Radon-Nikodym derivative) in Hilbert space employing Gaussian measure as reference - GP's.

Lecture.11. MCMC in Hilbert space : defining dimension invariant Markov transition kernel in Hilbert space and how overcomes degeneracy in high dimensions.

Lecture.12. Langevin Dynamics Simulation I - use of Langevin dynamics to simulate from a desired probability measure.

Lecture.13. Langevin Dynamics Simulation II  - use of approximate Langevin dynamics to simulate from a desired probability measure.

Lecture.14. Parallel Tempering - indtroduction to simulation from multi-modal target probability measures. 

Further notes

Machine Learning methods are having a major impact in every area of the engineering sciences. Machine Learning models and methods rely predominantly on Computational Statistics methods for model calibration, estimation, prediction and updating. Together Computational Statistics and Machine Learning are providing a revolution in the way mankind lives, works, communicates, and transacts.

Machine Learning methodology is not a magic wand that once waved will mysteriously solve long standing technical problems. There are underlying mathematical and statistical theories and principles which define these Machine Learning methods and it is important for the Machine Learning practitioner to have some understanding of them. This course is complementary to current Machine Learning modules in the Engineering Tripos.

This course will provide an overview and very basic introduction to a subset of the major theoretical and methodological ideas that underpin much of Machine Learning. It will provide the student with an appreciation of the possibilities and limitations of Machine Learning and Computational Statistics. This should be a launch pad for students wishing to gain a greater in-depth understanding of Machine Learning  as both practitioner and researcher.

Coursework

Coursework Format

Due date

& marks

Simulation Based Inference on Engineering Problem

The synthesis of both data and formal mathematical models in defining a digital twin of an engineering problem will be presented. The design of the machine learning and computational statistical methods to characterise uncertainty in predictions and forecasts from the digital twin will be the main focus of this exercise.

Learning objective:

  • To take an engineering problem and define appropriate mathematical, machine learning and data modelling strategies in studying the characteristics of the engineering system or artefact.
  • To successfully implement and deploy computational statistical methods in delivering an uncertainty quantification strategy in the specific engineering problem.

Individual Report

anonymously marked

  Wed week 9

[15/60]

 

Booklists

Shima, H. Functional Analysis for Physics and Engineering: An Introduction, CRC Press.

Biegler, L., Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallickj, B., Tenorio, L., van Bloemen Waanders, B., Willcox, K., and Marouk, Y. (2010). Large-Scale Inverse Problems and Quantification of Uncertainty. Wiley.

Brooks, S., Gelman, A., Jones, G. L., and Meng, X. (2011). Handbook of Markov Chain Monte Carlo. CRC.

Cotter, C. and Reich, S. (2015). Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press.

Law, K., Stuart, A., and Zygalakis, K. (2015). Data Assimilation: A Mathematical Introduction. Springer.

Rogers, S. and Girolami, M. (2016). A First Course in Machine Learning, 2nd Edition. CRC.

Sullivan, T. J. (2015). Introduction to Uncertainty Quantification. Springer.

 

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 04/06/2025 13:33

Engineering Tripos Part IA, Computer-Aided Design, 2025-26

Lecturer

Dr Richard Roebuck

Timing and Structure

This course involves: a single lecture in week 1 of Michaelmas Term; a Tutorials sheet to work though; a Tasks sheet on which you will be assessed. Help desk support is available through the term. Marking occurs at (or before) three fixed sessions.

Aims

The aims of the course are to:

  • Gain a working knowledge of Computer-aided Design (CAD) solid modelling.
  • Learn how to translate ideas, designs and real world items into shapes, assemblies and animations within a solid modelling environment.

Objectives

As specific objectives, by the end of the course students should be able to:

  • Use our chosen professional CAD package to create models of engineering components and assemblies.
  • Representing ideas, designs and real world items in the CAD environment in a range of ways.
  • Create output from the CAD environment, including animations, so as to be able to communicate ideas in a range of ways.

Content

The IA Computer-aided Design (CAD) course runs in Michaelmas Term and focusses on learning, and being assessed on, the operation of a professional CAD package.

The delivery of this course (lecture, helpdesks and marking sessions) are detailed on the moodle page supporting this course.

 

Michaelmas Term

  • Introduction to Solidworks
  • Creating parts
  • Forming assemblies
  • Outputting drawings
  • "Revolving"
  • "Sweeping"
  • Shape creation involving repeated "patterns"
  • Surface creation
  • Forming sheet metal objects
  • Using the "toolbox" of standard parts
  • Using "design tables"
  • Animating objects
  • Analysing the motion of animated objects

 

Further notes

There is a moodle page supporting the course. 

Examination Guidelines

Please refer to Form & conduct of the examinations.

 
Last modified: 05/06/2025 11:14

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