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Author: Boers, Niklas × Author: Kurths, Jürgen × End date: 2023 × Start date: 2020 × DDC: 530 × Type: article ×

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Now showing 1 - 3 of 3
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    Neural partial differential equations for chaotic systems
    ([London] : IOP, 2021) Gelbrecht, Maximilian; Boers, Niklas; Kurths, Jürgen
    When predicting complex systems one typically relies on differential equation which can often be incomplete, missing unknown influences or higher order effects. By augmenting the equations with artificial neural networks we can compensate these deficiencies. We show that this can be used to predict paradigmatic, high-dimensional chaotic partial differential equations even when only short and incomplete datasets are available. The forecast horizon for these high dimensional systems is about an order of magnitude larger than the length of the training data.
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    Reconstructing complex system dynamics from time series: a method comparison
    ([London] : IOP, 2020) Hassanibesheli, Forough; Boers, Niklas; Kurths, Jürgen
    Modeling complex systems with large numbers of degrees of freedom has become a grand challenge over the past decades. In many situations, only a few variables are actually observed in terms of measured time series, while the majority of variables - which potentially interact with the observed ones - remain hidden. A typical approach is then to focus on the comparably few observed, macroscopic variables, assuming that they determine the key dynamics of the system, while the remaining ones are represented by noise. This naturally leads to an approximate, inverse modeling of such systems in terms of stochastic differential equations (SDEs), with great potential for applications from biology to finance and Earth system dynamics. A well-known approach to retrieve such SDEs from small sets of observed time series is to reconstruct the drift and diffusion terms of a Langevin equation from the data-derived Kramers-Moyal (KM) coefficients. For systems where interactions between the observed and the unobserved variables are crucial, the Mori-Zwanzig formalism (MZ) allows to derive generalized Langevin equations that contain non-Markovian terms representing these interactions. In a similar spirit, the empirical model reduction (EMR) approach has more recently been introduced. In this work we attempt to reconstruct the dynamical equations of motion of both synthetical and real-world processes, by comparing these three approaches in terms of their capability to reconstruct the dynamics and statistics of the underlying systems. Through rigorous investigation of several synthetical and real-world systems, we confirm that the performance of the three methods strongly depends on the intrinsic dynamics of the system at hand. For instance, statistical properties of systems exhibiting weak history-dependence but strong state-dependence of the noise forcing, can be approximated better by the KM method than by the MZ and EMR approaches. In such situations, the KM method is of a considerable advantage since it can directly approximate the state-dependent noise. However, limitations of the KM approximation arise in cases where non-Markovian effects are crucial in the dynamics of the system. In these situations, our numerical results indicate that methods that take into account interactions between observed and unobserved variables in terms of non-Markovian closure terms (i.e., the MZ and EMR approaches), perform comparatively better. © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft
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    Complex systems approaches for Earth system data analysis
    (Bristol : IOP Publ., 2021) Boers, Niklas; Kurths, Jürgen; Marwan, Norbert
    Complex systems can, to a first approximation, be characterized by the fact that their dynamics emerging at the macroscopic level cannot be easily explained from the microscopic dynamics of the individual constituents of the system. This property of complex systems can be identified in virtually all natural systems surrounding us, but also in many social, economic, and technological systems. The defining characteristics of complex systems imply that their dynamics can often only be captured from the analysis of simulated or observed data. Here, we summarize recent advances in nonlinear data analysis of both simulated and real-world complex systems, with a focus on recurrence analysis for the investigation of individual or small sets of time series, and complex networks for the analysis of possibly very large, spatiotemporal datasets. We review and explain the recent success of these two key concepts of complexity science with an emphasis on applications for the analysis of geoscientific and in particular (palaeo-) climate data. In particular, we present several prominent examples where challenging problems in Earth system and climate science have been successfully addressed using recurrence analysis and complex networks. We outline several open questions for future lines of research in the direction of data-based complex system analysis, again with a focus on applications in the Earth sciences, and suggest possible combinations with suitable machine learning approaches. Beyond Earth system analysis, these methods have proven valuable also in many other scientific disciplines, such as neuroscience, physiology, epidemics, or engineering.