The intersection of dynamical systems and machine learning is largely unexplored but a start was made with a ‘Symposium on Machine Learning and Dynamical Systems’ in February 2019 at Imperial College London. This was followed by a ‘Second Symposium on Machine Learning and Dynamical Systems that was hosted online by the Fields Institute in September 2020.
The goal of this interest group is to bring together researchers from these fields to fill the gap between the theories of dynamical systems and machine learning, in the following directions:
Machine learning for dynamical systems
How to analyse dynamical systems on the basis of observed data rather than attempt to study models analytically.
Dynamical systems for machine learning
How to analyse algorithms of machine learning using tools from the theory of dynamical systems.
Explaining the science
Since its inception in the 19th century through the efforts of Poincaré and Lyapunov, the theory of dynamical systems addresses the qualitative behaviour of dynamical systems as understood from models. From this perspective, the modeling of dynamical processes in applications requires a detailed understanding of the processes to be analysed. This deep understanding leads to a model, which is an approximation of the observed reality and is often expressed by a system of ordinary/partial, underdetermined (control), deterministic/stochastic differential or difference equations. While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics, brain dynamics, biological systems or the financial markets), the development of such models is notably difficult.
On the other hand, the field of machine learning is concerned with algorithms designed to accomplish a certain task, whose performance improves with the input of more data. Applications for machine learning methods include computer vision, stock market analysis, speech recognition, recommender systems and sentiment analysis in social media. The machine learning approach is invaluable in settings where no explicit model is formulated, but measurement data is available. This is frequently the case in many systems of interest, and the development of data-driven technologies is becoming increasingly important in many applications.
The purpose of the interest group is to cross-fertilise between the two fields. Firstly, many machine learning algorithms are dynamical systems in their own right and dynamical systems insight can help understand whether they converge and to what, and to design better algorithms. Secondly, it is perfectly possible to incorporate partial mechanistic models into machine learning, and such a hybrid approach is highly worth developing further. Thirdly, there is great scope for dynamical systems theory to benefit from machine learning from the output of a system.
The research will also inform existing Turing programmes such as data-centric engineering, finance and economics, and health and medical sciences.
The interest group will also propose a Theory and Methods Challenge Fortnight.
How to get involved
Professor Weinan E, Princeton University
Aamal Hussain, Imperial College London
Professor Wei Kang, Department of Applied Mathematics, Naval Postgraduate School, Monterey, California, USA
Professor Erik M. Bollt Clarkson University, Potsdam, NY
Lyudmila Grigoryeva, University of Konstanz, Germany
Professor Juan-Pablo Ortega, NTU Singapore
Qianxiao Li National University of Singapore
Professor Sebastian van Strien, Imperial College London
Professor Marian Mrozek, Jagiellonian University in Kraków,
Professor Josef Teichmann, ETH Zürich
Konstantin Mischaikow, Rutgers University
Enrique Zuazua, Chair in Applied Analysis, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Azadeh Khaleghi, Lancaster University
Professor Jeroen Lamb, Imperial College London
Hedy Attouch, University of Montpellier