### Warwick Lecture

Academic Year 2021/2022

MA4J5-15 Mathematical Structures

of Complex Systems

### Overview

**Location: Warwick Mathematics Institute**

For examination support, look at the end of this page.

For examination support, look at the end of this page.

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.

### Administration

The day to day exchange of teaching material is done via a cloud application, to which course members will be granted access. Login URL:

https://001.uk.sara-computing.com/nextcloud/

The life teaching part of this lecture is done during Warwick term weeks 1-10, every Thursday 1 p.m. in room B3.02.

Most of the lecture material is also available online via Youtube at

SARA Systems Streaming

### 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.

Video link to Lecture 1

Video link to Lecture 2

Part I Structural Modelling

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.

Video to Lecture 4

**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.

Video to Lecture 6

Structural Modelling with Hierachies

**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.

Video link to Lecture 8

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

• Chemical reactions and Biochemistry.

• Covid-19 Epidemiology.

• Economics and Sociology, Agent-based modelling.

Video link to Lecture 9

Video link to Lecture 10

Video link to Lecture 16

**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.

Video link to Lecture 11

**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.

Video Link to Lecture 13

**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.

Examination Support:

Video Link