Publikationen

Prediction of multi-dimensional time-series data, which may represent such diverse phenomena as climate changes or financial markets, remains a challenging task in view of inherent nonlinearities and non-periodic behavior In contrast to other recurrent neural networks, echo state networks (ESNs) are attractive for (online) learning due to lower requirements w.r.t.training data and computational power. However, the randomly-generated reservoir renders the choice of suitable hyper-parameters as an open research topic. We systematically derive and exemplarily demonstrate design guidelines for the hyper-parameter optimization of ESNs. For the evaluation, we focus on the prediction of chaotic time series, an especially challenging problem in machine learning. Our findings demonstrate the power of a hyper-parameter-tuned ESN when auto-regressively predicting time series over several hundred steps. We found that ESNs’ performance improved by 85.1%-99.8% over an already wisely chosen default parameter initialization. In addition, the fluctuation range is considerably reduced such that significantly worse performance becomes very unlikely across random reservoir seeds. Moreover, we report individual findings per hyper-parameter partly contradicting common knowledge to further, help researchers when training new models.

Time-resolved photoluminescence (TRPL) measurements and the extraction of meaningful parameters involve four key ingredients: a suitable sample such as a semiconductor double heterostructure, a state-of-the-art measurement setup, a kinetic model appropriate for the description of the sample behavior, and a general analysis method to extract the model parameters of interest from the measured TRPL transients. Until now, the last ingredient is limited to single curve fits, which are mostly based on simple models and least-squares fits. These are often insufficient for the parameter extraction in real-world applications. The goal of this article is to give the community a universal method for the analysis of TRPL measurements, which accounts for the Poisson distribution of photon counting events. The method can be used to fit multiple TRPL transients simultaneously using general kinematic models, but should also be used for single transient fits. To demonstrate this approach, multiple TRPL transients of a GaAs/AlGaAs heterostructure are fitted simultaneously using coupled rate equations. It is shown that the simultaneous fits of several TRPL traces supplemented by systematic error estimations allow for a more meaningful and more robust parameter determination. The statistical methods also quantify the quality of the description by the underlying physical model.

A classical result by Rado characterises the so-called partition-regular matrices A, i.e. those matrices A for which any finite colouring of the positive integers yields a monochromatic solution to the equation Ax=0. We study the asymmetric random Rado problem for the (binomial) random set [n]p in which one seeks to determine the threshold for the property that any r-colouring, r≥2, of the random set has a colour i∈[r] admitting a solution for the matrical equation Aix=0, where A1,…,Ar are predetermined partition-regular matrices pre-assigned to the colours involved. We prove a 1-statement for the asymmetric random Rado property. In the symmetric setting our result retrieves the 1-statement of the symmetric random Rado theorem established in a combination of results by Rödl and Ruciânski [34] and by Friedgut, Rödl and Schacht [11]. We conjecture that our 1-statement in fact unveils the threshold for the asymmetric random Rado property, yielding a counterpart to the so-called Kohayakawa-Kreuter conjecture concerning the threshold for the asymmetric random Ramsey problem in graphs. We deduce the aforementioned 1-statement for the asymmetric random Rado property after establishing a broader result generalising the main theorem of Friedgut, Rödl and Schacht from [11]. The latter then serves as a combinatorial framework through which 1-statements for Ramsey-type problems in random sets and (hyper)graphs alike can be established in the asymmetric setting following a relatively short combinatorial examination of certain hypergraphs. To establish this framework we utilise a recent approach put forth by Mousset, Nenadov and Samotij [26] for the Kohayakawa-Kreuter conjecture.

Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples.

Extended Dynamic Mode Decomposition, embedded in the Koopman framework, is a widely-applied technique to predict the evolution of an observable along the flow of a dynamical (control) system. However, despite its popularity, the error analysis for control systems is still fragmentary. Here, we provide a complete and rigorous analysis of the approximation error for control systems. To this end, the approximation error is split up according to its two sources of error: the finite dictionary size (projection) and the finite amount of i.i.d. data used to generate the surrogate model (estimation). Then, invoking - among others - finite-elements techniques and the Chebyshev inequality, probabilistic error bounds are derived. Finally, we demonstrate the applicability of the novel error bounds in optimal control with state and control constraints.