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We present a novel, model-free, and data-driven methodology for controlling complex dynamical systems into previously unseen target states, including those with significantly different and complex dynamics. Leveraging a parameter-aware realization of next-generation reservoir computing, our approach accurately predicts system behavior in unobserved parameter regimes, enabling control over transitions to arbitrary target states. Crucially, this includes states with dynamics that differ fundamentally from known regimes, such as shifts from periodic to intermittent or chaotic behavior. The method's parameter-awareness facilitates non-stationary control, ensuring smooth transitions between states. By extending the applicability of machine learning-based control mechanisms to previously inaccessible target dynamics, this methodology opens the door to transformative new applications while maintaining exceptional efficiency. Our results highlight reservoir computing as a powerful alternative to traditional methods for dynamic system control.

Minkowski tensors are comprehensive shape descriptors that robustly capture n-point information in complex random geometries and that have already been extensively applied in the Euclidean plane. Here, we devise a novel framework for Minkowski tensors on the sphere. We first advance the theory by introducing irreducible Minkowski tensors, which avoid the redundancies of previous representations. We, moreover, generalize Minkowski sky maps to the sphere, i.e., a concept of local anisotropy, which easily adjusts to masked data. We demonstrate the power of our new procedure by applying it to simulations and real data of the Cosmic Microwave Background, finding an anomalous region close to the well-known Cold Spot. The accompanying open-source software, litchi, used to generate these maps from data in the HEALPix-format is made publicly available to facilitate broader integration of Minkowski maps in other fields, such as fluid demixing, porous structures, or geosciences more generally., Comment: Code available at https://github.com/ccollischon/litchi . Submitted to Communications Physics. 15 pages + 5 supplementary pages, 7 figures + 8 supplementary figures

Identifying and quantifying co-dependence between financial instruments is a key challenge for researchers and practitioners in the financial industry. Linear measures such as the Pearson correlation are still widely used today, although their limited explanatory power is well known. In this paper we present a much more general framework for assessing co-dependencies by identifying and interpreting linear and nonlinear causalities in the complex system of financial markets. To do so, we use two different causal inference methods, transfer entropy and convergent cross-mapping, and employ Fourier transform surrogates to separate their linear and nonlinear contributions. We find that stock indices in Germany and the U.S. exhibit a significant degree of nonlinear causality and that correlation, while a very good proxy for linear causality, disregards nonlinear effects and hence underestimates causality itself. The presented framework enables the measurement of nonlinear causality, the correlation-causality fallacy, and motivates how causality can be used for inferring market signals, pair trading, and risk management of portfolios. Our results suggest that linear and nonlinear causality can be used as early warning indicators of abnormal market behavior, allowing for better trading strategies and risk management., Comment: 14 pages, 10 figures

Model-free and data-driven prediction of tipping point transitions in nonlinear dynamical systems is a challenging and outstanding task in complex systems science. We propose a novel, fully data-driven machine learning algorithm based on next-generation reservoir computing to extrapolate the bifurcation behavior of nonlinear dynamical systems using stationary training data samples. We show that this method can extrapolate tipping point transitions. Furthermore, it is demonstrated that the trained next-generation reservoir computing architecture can be used to predict non-stationary dynamics with time-varying bifurcation parameters. In doing so, post-tipping point dynamics of unseen parameter regions can be simulated.

Early warning systems are an essential tool for effective humanitarian action. Advance warnings on impending disasters facilitate timely and targeted response which help save lives and livelihoods. In this work we present a quantitative methodology to forecast levels of food consumption for 60 consecutive days, at the sub-national level, in four countries: Mali, Nigeria, Syria, and Yemen. The methodology is built on publicly available data from the World Food Programme's global hunger monitoring system which collects, processes, and displays daily updates on key food security metrics, conflict, weather events, and other drivers of food insecurity. In this study we assessed the performance of various models including Autoregressive Integrated Moving Average (ARIMA), Extreme Gradient Boosting (XGBoost), Long Short Term Memory (LSTM) Network, Convolutional Neural Network (CNN), and Reservoir Computing (RC), by comparing their Root Mean Squared Error (RMSE) metrics. Our findings highlight Reservoir Computing as a particularly well-suited model in the field of food security given both its notable resistance to over-fitting on limited data samples and its efficient training capabilities. The methodology we introduce establishes the groundwork for a global, data-driven early warning system designed to anticipate and detect food insecurity., Comment: 19 pages, 7 figures + supplementary material

We investigate the stationary (late-time) training regime of single- and two-layer underparameterized linear neural networks within the continuum limit of stochastic gradient descent (SGD) for synthetic Gaussian data. In the case of a single-layer network in the weakly underparameterized regime, the spectrum of the noise covariance matrix deviates notably from the Hessian, which can be attributed to the broken detailed balance of SGD dynamics. The weight fluctuations are in this case generally anisotropic, but effectively experience an isotropic loss. For an underparameterized two-layer network, we describe the stochastic dynamics of the weights in each layer and analyze the associated stationary covariances. We identify the inter-layer coupling as a distinct source of anisotropy for the weight fluctuations. In contrast to the single-layer case, the weight fluctuations are effectively subject to an anisotropic loss, the flatness of which is inversely related to the fluctuation variance. We thereby provide an analytical derivation of the recently observed inverse variance-flatness relation in a model of a deep linear neural network., Comment: 27 pages, 9 figures

Controlling nonlinear dynamical systems using machine learning allows to not only drive systems into simple behavior like periodicity but also to more complex arbitrary dynamics. For this, it is crucial that a machine learning system can be trained to reproduce the target dynamics sufficiently well. On the example of forcing a chaotic parametrization of the Lorenz system into intermittent dynamics, we show first that classical reservoir computing excels at this task. In a next step, we compare those results based on different amounts of training data to an alternative setup, where next-generation reservoir computing is used instead. It turns out that while delivering comparable performance for usual amounts of training data, next-generation RC significantly outperforms in situations where only very limited data is available. This opens even further practical control applications in real world problems where data is restricted., Comment: IJCNN 2023

Multifunctional neural networks are capable of performing more than one task without changing any network connections. In this paper we explore the performance of a continuous-time, leaky-integrator, and next-generation `reservoir computer' (RC), when trained on tasks which test the limits of multifunctionality. In the first task we train each RC to reconstruct a coexistence of chaotic attractors from different dynamical systems. By moving the data describing these attractors closer together, we find that the extent to which each RC can reconstruct both attractors diminishes as they begin to overlap in state space. In order to provide a greater understanding of this inhibiting effect, in the second task we train each RC to reconstruct a coexistence of two circular orbits which differ only in the direction of rotation. We examine the critical effects that certain parameters can have in each RC to achieve multifunctionality in this extreme case of completely overlapping training data., Comment: Accepted for publication in the proceedings of the 2022 International Joint Conference on Neural Networks (IJCNN 2022)

Recently, Minkowski Tensors (MT) have gained popularity for morphological analysis tasks. As opposed to the scalar Minkowski functionals (MF; in 2D given by area, perimeter and Euler characteristic), MT can characterize symmetry and orientation of a body. This has been used for a variety of tasks, e.g. to detect interstellar bubbles by tracing back the origins of filaments in HII-regions, or to search for alignment of structures in the CMB. I present a marching-square-based method for calculating MT and MF on the sphere for maps in the Healpix format. MT are calculated for a local neighborhood and can then be summed up/averaged over a larger region, using their additivity property. This provides the possibility of localized analyses looking for CMB anisotropies and non-Gaussianities at varying scales., Comment: Contribution to the 2022 Cosmology session of the 56th Rencontres de Moriond

We investigated the self-excited as well as optogalvanically stimulated heartbeat instability in RF discharge complex plasma. Three video cameras measured the motion of the microparticles, the plasma emission, and the laser-induced fluorescence simultaneously. Comprehensive studies of the optogalvanic control of the heartbeat instability revealed that the microparticle suspension can be stabilized by a continuous laser, whereas a modulated laser beam induces the void contraction either transiently or resonantly. The resonance occurred when the laser modulation frequency coincided with the frequency of small breathing oscillations of the microparticle suspension, which are known to be a prerequisite to the heartbeat instability. Based on the experimental results we suggest that the void contraction during the instability is caused by an abrupt void transition from the dim to the bright regime [Pikalev et al., Plasma Sources Sci. Technol. 30, 035014 (2021)]. In the bright regime, a time-averaged electric field at the void boundary heats the electrons causing bright plasma emission inside the void. The dim void has much lower electric field at the boundary and exhibits therefore no emission feature associated with it., Comment: 10 pages, 10 figures, supplemental materials

We generalize the concept of local states (LS) for the prediction of high-dimensional, potentially mixed chaotic systems. The construction of generalized local states (GLS) relies on defining distances between time series on the basis of their (non-)linear correlations. We demonstrate the prediction capabilities of our approach based on the reservoir computing (RC) paradigm using the Kuramoto-Sivashinsky (KS), the Lorenz-96 (L96) and a combination of both systems. In the mixed system a separation of the time series belonging to the two different systems is made possible with GLS. More importantly, prediction remains possible with GLS, where the LS approach must naturally fail. Applications for the prediction of very heterogeneous time series with GLSs are briefly outlined., Comment: 9 pages, 10 figures, published by Physical Review Research

We propose a novel and fully data driven control scheme which relies on machine learning (ML). Exploiting recently developed ML-based prediction capabilities of complex systems, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline our approach using the examples of the Lorenz and the R\"ossler system and show how these systems can very accurately be brought not only to periodic but also to e.g. intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications that range from engineering to medicine., Comment: 5 Pages, 4 Figures, Supplemental Material Attached

Reservoir computing has repeatedly been shown to be extremely successful in the prediction of nonlinear time-series. However, there is no complete understanding of the proper design of a reservoir yet. We find that the simplest popular setup has a harmful symmetry, which leads to the prediction of what we call mirror-attractor. We prove this analytically. Similar problems can arise in a general context, and we use them to explain the success or failure of some designs. The symmetry is a direct consequence of the hyperbolic tangent activation function. Further, four ways to break the symmetry are compared numerically: A bias in the output, a shift in the input, a quadratic term in the readout, and a mixture of even and odd activation functions. Firstly, we test their susceptibility to the mirror-attractor. Secondly, we evaluate their performance on the task of predicting Lorenz data with the mean shifted to zero. The short-time prediction is measured with the forecast horizon while the largest Lyapunov exponent and the correlation dimension are used to represent the climate. Finally, the same analysis is repeated on a combined dataset of the Lorenz attractor and the Halvorsen attractor, which we designed to reveal potential problems with symmetry. We find that all methods except the output bias are able to fully break the symmetry with input shift and quadratic readout performing the best overall., Comment: 14 pages, 10 figures, accepted by chaos

Reservoir computing is a very promising approach for the prediction of complex nonlinear dynamical systems. Besides capturing the exact short-term trajectories of nonlinear systems, it has also proved to reproduce its characteristic long-term properties very accurately. However, predictions do not always work equivalently well. It has been shown that both short- and long-term predictions vary significantly among different random realizations of the reservoir. In order to gain an understanding on when reservoir computing works best, we investigate some differential properties of the respective realization of the reservoir in a systematic way. We find that removing nodes that correspond to the largest weights in the output regression matrix reduces outliers and improves overall prediction quality. Moreover, this allows to effectively reduce the network size and, therefore, increase computational efficiency. In addition, we use a nonlinear scaling factor in the hyperbolic tangent of the activation function. This adjusts the response of the activation function to the range of values of the input variables of the nodes. As a consequence, this reduces the number of outliers significantly and increases both the short- and long-term prediction quality for the nonlinear systems investigated in this study. Our results demonstrate that a large optimization potential lies in the systematical refinement of the differential reservoir properties for a given dataset., Comment: 11 pages, 8 figures, published in Chaos

The prediction of complex nonlinear dynamical systems with the help of machine learning techniques has become increasingly popular. In particular, reservoir computing turned out to be a very promising approach especially for the reproduction of the long-term properties of a nonlinear system. Yet, a thorough statistical analysis of the forecast results is missing. Using the Lorenz and R\"ossler system we statistically analyze the quality of prediction for different parametrizations - both the exact short-term prediction as well as the reproduction of the long-term properties (the "climate") of the system as estimated by the correlation dimension and largest Lyapunov exponent. We find that both short and longterm predictions vary significantly among the realizations. Thus special care must be taken in selecting the good predictions as predictions which deliver better short-term prediction also tend to better resemble the long-term climate of the system. Instead of only using purely random Erd\"os-Renyi networks we also investigate the benefit of alternative network topologies such as small world or scale-free networks and show which effect they have on the prediction quality. Our results suggest that the overall performance with respect to the reproduction of the climate of both the Lorenz and R\"ossler system is worst for scale-free networks. For the Lorenz system there seems to be a slight benefit of using small world networks while for the R\"ossler system small world and Erd\"os -Renyi networks performed equivalently well. In general the observation is that reservoir computing works for all network topologies investigated here., Comment: 10 pages, 5 figures, accepted for publication in Chaos

The Fourier phase information play a key role for the quantified description of nonlinear data. We present a novel tool for time series analysis that identifies nonlinearities by sensitively detecting correlations among the Fourier phases. The method, being called phase walk analysis, is based on well established measures from random walk analysis, which are now applied to the unwrapped Fourier phases of time series. We provide an analytical description of its functionality and demonstrate its capabilities on systematically controlled leptokurtic noise. Hereby, we investigate the properties of leptokurtic time series and their influence on the Fourier phases of time series. The phase walk analysis is applied to measured and simulated intermittent time series, whose probability density distribution are approximated by power laws. We use the day-to-day returns of the Dow-Jones industrial average, a synthetic time series with tailored nonlinearities mimicing the power law behavior of the Dow-Jones and the acceleration of wind at an Atlantic offshore site. Testing for nonlinearities by means of surrogates shows that the new method yields strong significances for nonlinear behavior. Due to the drastically decreased computing time as compared to embedding space methods the number of surrogate realizations can be increased by orders of magnitude. Thereby, the probability distribution of the test statistics can very accurately be derived and parametrized which allows for much more precise tests on nonlinearities., Comment: 9 pages, 7 figures, to be published in Chaos, Volume 28, Issue 6, June 2018

Pearson correlation and mutual information based complex networks of the day-to-day returns of US S&P500 stocks between 1985 and 2015 have been constructed in order to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies is derived using surrogate data and leads to interesting observations during periods of financial market crises. In contrast to the expectation that dependencies reduce mainly to linear correlations during crises we show that (at least in the 2008 crisis) nonlinear effects are significantly increasing. It turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we demonstrate with the example of the 2008 subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio., Comment: 12 pages, 11 figures, Phys. Rev. E, accepted

We study a complex plasma under microgravity conditions that is first stabilized with an oscillating electric field. Once the stabilization is stopped, the so-called heartbeat instability develops. We study how the kinetic energy spectrum changes during and after the onset of the instability and compare with the double cascade predicted by Kraichnan and Leith for two-dimensional turbulence. The onset of the instability manifests clearly in the ratio of the reduced rates of cascade of energy and enstrophy and in the power-law exponents of the energy spectra., Comment: 7 pages, 7 figures

The wake-mediated propulsion of an "extra" particle in a channel of two neighboring rows of a two-dimensional plasma crystal, observed experimentally by Du et al. [Phys. Rev. E 89, 021101(R) (2014)], is explained in simulations and theory. We use the simple model of a pointlike ion wake charge to reproduce this intriguing effect in simulations, allowing for a detailed investigation and a deeper understanding of the underlying dynamics. We show that the nonreciprocity of the particle interaction, owing to the wake charges, is responsible for a broken symmetry of the channel that enables a persistent self-propelled motion of the extra particle. We find good agreement of the terminal extra-particle velocity with our theoretical considerations and with experiments., Comment: 7 pages, 4 figures, PRL (https://journals.aps.org/prl/), updated version with correct author affiliations

The performance of recurrence networks and symbolic networks to detect weak nonlinearities in time series is compared to the nonlinear prediction error. For the synthetic data of the Lorenz system, the network measures show a comparable performance. In the case of relatively short and noisy real-world data from active galactic nuclei, the nonlinear prediction error yields more robust results than the network measures. The tests are based on surrogate data sets. The correlations in the Fourier phases of data sets from some surrogate generating algorithms are also examined. The phase correlations are shown to have an impact on the performance of the tests for nonlinearity., Comment: 9 pages, 5 figures, Chaos (http://scitation.aip.org/content/aip/journal/chaos), corrected typos

A molecular dynamics simulation of the demixing process of a binary complex plasma is analysed and the role of distinct interaction potentials is discussed by using morphological Minkowski tensor analysis of the minority phase domain growth in a demixing simulated binary complex plasma. These Minkowski tensor methods are compared with previous results that utilized a power spectrum method based on the time-dependent average structure factor. It is shown that the Minkowski tensor methods are superior to the previously used power spectrum method in the sense of higher sensitivity to changes in domain size. By analysis of the slope of the temporal evolution of Minkowski tensor measures qualitative differences between the case of particle interaction with a single length scale compared to particle interactions with two different length scales (dominating long range interaction) are revealed. After proper scaling the graphs for the two length scale scenario coincide, pointing towards universal behaviour. The qualitative difference in demixing scenarios is evidenced by distinct demixing behaviour: In the long range dominated cases demixing occurs in two stages. At first neighbouring particles agglomerate then domains start to merge in cascades. However in the case of only one interaction length scale only agglomeration but no merging of domains can be observed. Thus, Minkowski Tensor analysis are likely to become a useful tool for further investigation of this (and other) demixing processes. It is capable to reveal (nonlinear) local topological properties, probing deeper than (linear) global power spectrum analysis, however still providing easily interpretable results founded on a solid mathematical framework., Comment: 12 pages, 10 figures, Phys. Rev. E, accepted for publication, http://journals.aps.org/pre

The spectral asymmetry of the wave energy distribution of dust particles during mode-coupling induced melting, observed for the first time in plasma crystals by Cou\"edel et al. [Phys. Rev. E 89, 053108 (2014)], is studied theoretically and by molecular-dynamics simulations. It is shown that an anisotropy of the well confining the microparticles selects the directions of preferred particle motion. The observed differences in intensity of waves of opposed directions is explained by a nonvanishing phonon flux. Anisotropic phonon scattering by defects and Umklapp scattering are proposed as possible reasons for the mean phonon flux., Comment: 12 pages, 8 figures, Phys. Rev. E (http://journals.aps.org/pre), accepted

Network analysis was used to study the structure and time evolution of driven three-dimensional complex plasma clusters. The clusters were created by suspending micron-size particles in a glass box placed on top of the rf electrode in a capacitively coupled discharge. The particles were highly charged and manipulated by an external electric field that had a constant magnitude and uniformly rotated in the horizontal plane. Depending on the frequency of the applied electric field, the clusters rotated in the direction of the electric field or remained stationary. The positions of all particles were measured using stereoscopic digital in-line holography. The network analysis revealed the interplay between two competing symmetries in the cluster. The rotating cluster was shown to be more cylindrical than the nonrotating cluster. The emergence of vertical strings of particles was also confirmed., Comment: 9 pages, 9 figures; corrected Fig.4 and typos

We study the onset and characteristics of vortices in complex (dusty) plasmas using two-dimensional simulations in a setup modeled after the PK-3 Plus laboratory. A small number of microparticles initially self-arranges in a monolayer around the void. As additional particles are introduced, an extended system of vortices develops due to a non-zero curl of the plasma forces. We demonstrate a shear-thinning effect in the vortices. Velocity structure functions and the energy and enstrophy spectra show that vortex flow turbulence is present that is in essence of the "classical" Kolmogorov type., Comment: 5 pages, 6 figures

How did the universe evolve? The fine angular scale (l>1000) temperature and polarization anisotropies in the CMB are a Rosetta stone for understanding the evolution of the universe. Through detailed measurements one may address everything from the physics of the birth of the universe to the history of star formation and the process by which galaxies formed. One may in addition track the evolution of the dark energy and discover the net neutrino mass. We are at the dawn of a new era in which hundreds of square degrees of sky can be mapped with arcminute resolution and sensitivities measured in microKelvin. Acquiring these data requires the use of special purpose telescopes such as the Atacama Cosmology Telescope (ACT), located in Chile, and the South Pole Telescope (SPT). These new telescopes are outfitted with a new generation of custom mm-wave kilo-pixel arrays. Additional instruments are in the planning stages., Comment: Science White Paper submitted to the US Astro2010 Decadal Survey. Full list of 177 author available at http://cmbpol.uchicago.edu

The method of surrogates is widely used in the field of nonlinear data analysis for testing for weak nonlinearities. The two most commonly used algorithms for generating surrogates are the amplitude adjusted Fourier transform (AAFT) and the iterated amplitude adjusted Fourier transfom (IAAFT) algorithm. Both the AAFT and IAAFT algorithm conserve the amplitude distribution in real space and reproduce the power spectrum (PS) of the original data set very accurately. The basic assumption in both algorithms is that higher-order correlations can be wiped out using a Fourier phase randomization procedure. In both cases, however, the randomness of the Fourier phases is only imposed before the (first) Fourier back tranformation. Until now, it has not been studied how the subsequent remapping and iteration steps may affect the randomness of the phases. Using the Lorenz system as an example, we show that both algorithms may create surrogate realizations containing Fourier phase correlations. We present two new iterative surrogate data generating methods being able to control the randomization of Fourier phases at every iteration step. The resulting surrogate realizations which are truly linear by construction display all properties needed for surrogate data., Comment: To appear in: "Topics on Chaotic Systems: Selected Papers from Chaos 2008 International Conference", World Scientific

We present a methodology for detecting non-linearities in data sets based on the characterization of the structural features of the Fourier phase maps. A Fourier phase map is a 2D set of points $M= \{(\phi_{\vec{k}}, \phi_{\vec{k} + \vec{\Delta}})\}$, where $ \phi_{\vec{k}}$ is the phase of the $k$-mode of the Fourier transform of the data set and $\vec{\Delta}$ a phase shift. The information thus rendered on this space is analyzed using the spectrum of weighted scaling indices to detect phase coupling at any scale $\vec{\Delta}$. We propose a statistical test of significance based on the comparison of the properties of phase maps created from both the original data and surrogate realizations. We have applied our method to the Lorenz system and the logarithmic stock returns of the Dow Jones index. Applications to higher dimensional data are straightforward. The results indicate that both the Lorenz system and the Dow Jones time series exhibit significant signatures of non-linear behavior., Comment: 4 pages, 4 figures