Introductory description

We keep also this year the partition of the lecture into three parts, structural modelling, dynamic modelling and learning/data analysis. All of these parts have proven to be necessary for any complex systems modelling, sich as models in the Life Sciences, in the Social Sciences, in Economy & Finance or Ecology and Infectious Diseases.

In this lecture will learn how to start the modelling process by thinking about the model's static structure, which then in a dynamic model gives rise to the choice of variables. Finally, with the dive into mathematical learning theories, the students will understand that a mathematical model is never finished, but needs recursive learning steps to improve its parametrisation and even structure.

A very important aspect of the lecture is the smooth transition from static to dynamic stochastic models with the help of rule-based system descriptions which have evolved from the modelling of chemical reactions.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Weekly Overview

Introduction:

Week 1: Mathematical Modelling, Past, Present and Future

• What is Mathematical Modelling?
• Why Complex Systems?..
• Philosophy of Science, Empirical Data and Prediction.
• About this course.

Part I Structural Modelling

Week 2: Relational Structures

• Relational family: hypergraphs, simplicial complexes and hierachical hypergraphs.
• Graph characteristics, examples from real world complex systems (social science, infrastructure, economy, biology, internet).
• Introduction to algebraic and computational graph theory.

Week 3: Transformations of Relational Models

• Connections between graphs, hypergraphs, simplicial complexes and hierachical hypergraphs.
• Applications of hierachical hypergraphs.
• Stochastic processes of changing relational model topologies.

Part II Dynamic Modelling

Week 4: Stochastic Processes

• Basic concepts, Poisson Process.
• Opinion formation: relations and correlations.
• Master eqation type-rule based stochastic collision processes.

Week 5: Applications of type-rule based stochastic collision processes

• Chemical reactions and Biochemistry.
• Covid-19 Epidemiology.
• Economics and Sociology, Agent-based modelling.

Week 6: Dynamical Systems (single compartment)

• Basic concepts, examples.
• Relation between type-rule-based stochastic collision processes in single compartments and ODE
• Applications, connections between dynamical systems and structural modelling (from Part I), the interaction graph, feedback loops.
• Time scales: evolutionary outlook.

Week 7: Spatial processes and Partial Differential Equations:

• Type-rule-based multi-compartment models.
• Reaction-Diffusion Equations.
• Applications.

Part III Data Analysis and Machine Learning

Week 8: Statistics and Mathematical Modelling

• Statistical Models and Data.
• Classification.
• Parametrisation.

Week 9: Machine Learning and Mathematical Modelling:

• Mathematical Learning Theory.
• Bayesian Networks.
• Bayesian Model Selection.

Week 10: Neural Networks and Deep Learning:

• Basic concepts.
• Neural Networks and Machine Learning.
• Discussion and outlook.

Learning outcomes

By the end of the module, students should be able to:

• Know basic mathematical modelling techniques to describe different complex systems.
• Choose a set of mathematical methods appropriate to tackle and investigate models of complex systems.
• Develop research interest or practical skills to solve real-world problems related to complex systems, especially with the help of computational methods.
• Know some ideas how mathematical techniques to investigate complex systems should or could be developed further.

Interdisciplinary

This module is highly interdisciplinary, as is the nature of mathematical modelling lectures. Mathematical Modelling is applied to complex systems, which can be in nearly any scientific area, and combinations of scientific areas. Examples are Biology, Chemistry, Biochemistry, Medicine, Social Sciences, Economy and Finance.

International

As we like to move more towards online lectures, we like to offer this module to Warwick teaching partners online.

Subject specific skills

See learning outcomes.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.