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Machine Learning for Computational Dynamics

Periodic Reporting for period 1 - MALCOD (Machine Learning for Computational Dynamics)

Reporting period: 2015-09-01 to 2017-08-31

The project aimed to establish new methods for the numerical analysis of dynamical systems by using tools from the field of machine learning. This interdisciplinary work was largely unexplored, the MALCOD project began a systematic development of this field. The approach was motivated by kernel methods in machine learning, in particular the functional analytic framework provided by the reproducing kernel Hilbert space (RKHS) that underlies many modern machine learning methods.

The key objectives for the research outlined in the proposal were:

1. Numerical approximation and continuation for control and random dynamical systems
2. Numerical approximation of Lyapunov functions and basins of attraction for deterministic dynamical systems
3. Error analysis of the numerical methods from 1 and 2
4. Application to problems in power grid networks, movie image rendering and turbulent flow across aerofoils

The project work was based at the Potsdam Institute for Climate Impact Research (PIK). Another key component of the project was to have secondments with the non-academic partner organisation Ambrosys GmbH in Potsdam, Germany.

The project was terminated early, and ran from 01.09.2015 to 31.03.2016.
In the time that the project ran for, there were several achievements made. In the beginning of the project, the fellow presented his own research and project to the group at PIK. This lead to regular discussions with several members of the group on machine learning methods and dynamical systems theory, ultimately leading to two publications. A further two publications were finalised within the project timespan. Research results were presented at the DMV conference in Hamburg in September 2015. In addition, I spent the time from 22.02.16 to 27.03.16 in secondment at Ambrosys GmbH as planned. During the secondment, work was commenced on applying machine learning methods to the problem of audio synthesis, and the fellow received industrial training The following is the list of publications that were either finalised or submitted for publication during the project timespan:

1. M. Rasmussen, J. Rieger and K. N. Webster, “Approximation of reachable sets using optimal control and support vector machines”, Journal of Computational and Applied Mathematics, 311 (2017), 68-83.
2. P. Giesl, B. Hamzi, M. Rasmussen and K. N. Webster, “Approximation of Lyapunov functions from data”, submitted to Journal of Computational and Applied Mathematics.
3. T. Kittel, J. Heitzig, K. N. Webster and J. Kurths, “Timing of transients”, New Journal of Physics, to appear.
4. P. Schultz, F. Hellmann, K. N. Webster and J. Kurths, “Finite-time basin stability and independence times”, Chaos, under review.
In the limited timespan for which the project ran for, the results exceeded the initial expectations contained in the proposal. There have been four publications, a presentation at a major conference, and significant transfer of knowledge and learning between the fellow, the host institution and the secondment partner organisation.

The work achieved in combining machine learning methods with numerical analysis problems was highly original work, and paved the way for future research to be conducted along similar lines. The field of machine learning in particular is a fast evolving field, and many breakthroughs have since been made in the areas of Gaussian processes and deep learning, and there will be plenty of scope for continuing this programme by exploiting and modifying these newer algorithms.

In addition, dissemination and communication of research findings has lead to an immediate impact for the fellow’s research career and industrial impact. It has also contributed towards enhancing European excellence in applied research.
Numerical Estimation of Regularized Reaching Time as a Lyapunov-like function with constant negative