Machine Learning for Nonlinear Dynamical Systems

Description

Nonlinear dynamical systems arise in many scientific domains, including weather forecasting, energy systems, fluid dynamics, ecology, and other complex physical systems. These systems are traditionally modelled using mathematical or physical equations, but in practice such models are often imperfect. Model errors, uncertain parameters, incomplete observations, and uncertainty in initial conditions can all limit the reliability of forecasts.

The aim of this project is to explore how machine learning methods can be used to aid the modelling and prediction of nonlinear dynamical systems. Students will investigate how data-driven approaches can complement or replace traditional physical models, with a focus on short-term prediction, model error correction, and efficient approximation of system behaviour.

Group Project

The group project will serve as an introduction to nonlinear dynamical systems and basic machine learning approaches for time series modelling. Students will work together to understand simple dynamical systems, generate simulated data, and apply introductory machine learning methods to prediction problems.

By the end of the group project, students will have learned:

• The basic structure of nonlinear dynamical systems, such as discrete-time maps or simple ordinary differential equation systems.
• How to simulate simple dynamical systems numerically.
• How uncertainty in initial conditions and parameters can affect model predictions.
• The basic principles of machine learning models for time series prediction, such as regression methods, tree-based models, and simple neural networks.
• How to evaluate and compare predictive performance using appropriate error measures.

By the end of the group project, students will be able to:

• Implement simple nonlinear dynamical systems.
• Generate and analyse simulated time series data.
• Train basic machine learning models for short-term prediction.
• Compare data-driven models with physical models.
• Interpret numerical results and discuss the limitations of different modelling approaches.

Individual Project

The individual project will build on the work completed in the group project. Students will be expected to show additional independence by identifying and developing a suitable extension. The individual component may focus on a more advanced model, a more detailed numerical experiment, a comparison of different methods, or an application to a particular dataset.

Possible individual directions include:

• Using more advanced machine learning methods, such as deep neural networks, recurrent neural networks, or transformer-style models, for time series prediction.
• Using machine learning to correct systematic errors in an imperfect physical model.
• Studying sensitivity to initial conditions and uncertainty in predictions.
• Comparing purely data-driven models with hybrid or physics-informed approaches.
• Applying the methods to real or semi-real datasets, such as weather, energy demand, environmental monitoring, or other time series data.
• Exploring probabilistic forecasting and uncertainty quantification for nonlinear dynamical systems.
• Investigating how well a trained model generalises when the system parameters or data-generating regime change.

Mode of Operation and Evidence of Learning

This project develops understanding of nonlinear dynamical systems and their data-driven modelling through a combination of reading, discussion, and computational implementation. The emphasis is on linking mathematical structure, algorithmic methods, and empirical behaviour.

Students will:

• Read and synthesise material on dynamical systems and machine learning approaches for time series.
• Implement models in Python/R, including both physical dynamical systems and data-driven models.
• Compare theoretical properties with behaviour observed in simulated and real data.
• Analyse model performance and limitations under different assumptions.
• Critically evaluate modelling choices and their implications.

Understanding will be demonstrated through the ability to move between mathematical formulation, computational implementation, and interpretation of results. Evidence of learning will include code, written reports, and presentations, with emphasis on clear reasoning and critical analysis.

Prerequisites

Students should have some background in statistical modelling and basic programming.

• Statistical Inference II, for familiarity with standard statistical ideas and experience with R.
• Statistical Modelling II, helpful for understanding simple statistical models.

Resources

• An Introduction to Statistical Learning: https://www.statlearning.com/
• Probabilistic Machine Learning: An Introduction: https://probml.github.io/pml-book/book1.html
• Additional notes and reading materials will be provided during the project.

Areas of Research

Machine learning, nonlinear dynamical systems, time series modelling, uncertainty quantification, probabilistic forecasting, applied mathematics, and data science.