Periodic Reporting for period 1 - STALDYS (Statistical Learning for Dynamical Systems) Reporting period: 2019-02-01 to 2021-01-31 Summary of the context and overall objectives of the project a. What is the problem/issue being addressed? 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 proposal is to cross-fertilise between the two fields and particularly by using kernel methods for dynamical systems.b. Why is it important for society? Due to the prevalence of machine learning techniques in many fields and lack of understanding of why they work, approaches based on the theory of dynamical systems will definitely help understanding machine learning techniques and delineate their domain of applicability.c. What are the overall objectives? The overall objectives is to use kernel methods for the analysis of nonlinear dynamical systems. These methods are mathematically well grounded and using them for dynamical systems is a good way of having sound mathematical results for machine learning approaches to complex systems. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far a. Organized two symposia on Machine Learning and Dynamical Systems. The first one was at Imperial College London in Feb. 2019, https://sites.google.com/site/boumedienehamzi/symposium-on-machine-learning-for-dynamical-systems_2019 and the second one was online at the Fields institute, http://www.fields.utoronto.ca/activities/20-21/dynamical - b. I published several articles• Boumediene Hamzi, Romit Maulik and Houman Owhadi (2021). Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods, submitted, https://arxiv.org/abs/2103.10935.• F. Colonius and B. Hamzi (2020). Entropy for practical stabilization, accepted for publication in SIAM Journal of Control and Optimization, https://arxiv.org/abs/2009.08187• B. Haasdonk, B. Hamzi, G. Santin and D. Wittwar (2020). Kernel Methods for Center Manifold Approximation and a Data-Based Version of the Center Manifold Theorem, accepted for publication in Physica D, https://arxiv.org/abs/2012.00338• Bittracher, S. Klus, B. Hamzi, C Sch\"utte (2021), A kernel-based method for coarse graining complex dynamical systems, Journal of Nonlinear Science, https://arxiv.org/abs/1904.08622• Hamzi and H. Owhadi (2020). Learning dynamical systems from data: a simple cross-validation perspective, Physica D, https://arxiv.org/abs/2007.05074• Haasdonk, B. Hamzi, G. Santin and D. Wittwar (2020). Greedy Kernel Methods for Center Manifold Approximation, in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Springer Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 134), pp. 95--106, https://arxiv.org/abs/1810.11329• S. Klus, F. N\"uske and B. Hamzi (2020). Kernel-based approximation of the Koopman generator and Schr\"odinger operator, Entropy, 22, 722, https://arxiv.org/abs/2005.13231c. I edited a special issue in Physica D on “Machine Learning and Dynamical Systems” https://www.sciencedirect.com/journal/physica-d-nonlinear-phenomena/special-issue/100SM4JKVTDd. I started organizing a Special Interest Group on Machine Learning and Dynamical Systems at the Alan Turing Institute, https://www.turing.ac.uk/research/interest-groups/machine-learning-and-dynamical-systems e. I supervised MSc. Students at Imperial College London.f. I gave several talks.g. I created a Youtube Channel for online seminars on MLDS, https://www.youtube.com/channel/UCetvKhuAbnuU1tidgCz9g0g/videos Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) Extending the method of kernel flows to predicting dynamical systems, allowed to beat state of the art methods in climate modelling, cf. Boumediene Hamzi, Romit Maulik and Houman Owhadi (2021). Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods, submitted, https://arxiv.org/abs/2103.10935. stalyds.jpg
