@unpublished{64865,
abstract = {{We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.}},
author = {{von der Gracht, Sören and Lohse, Alexander}},
booktitle = {{arXiv:2603.06157}},
title = {{{Design of Hierarchical Excitable Networks}}},
year = {{2026}},
}
@article{64979,
abstract = {{We investigate homogeneous coupled cell systems with high-dimensional internal dynamics. In many studies on network dynamics, the analysis is restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between dynamical behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the underlying network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional internal dynamics case. This has interesting implications for possible center subspaces in bifurcation analysis. We describe the effect on steady state and Hopf bifurcations in l-parameter families of network vector fields. The main results in that regard are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least l-dimensional for steady state bifurcations and 2l-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain even greater understanding of bifurcation scenarios in the high-dimensional case beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). As another main result, we prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.}},
author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
issn = {{0960-0779}},
journal = {{Chaos, Solitons & Fractals}},
keywords = {{Coupled cell systems, Network dynamics, Dimension reduction, Bifurcation theory, Symmetry, Monoid representation theory}},
publisher = {{Elsevier BV}},
title = {{{Homogeneous coupled cell systems with high-dimensional internal dynamics}}},
doi = {{10.1016/j.chaos.2026.118196}},
volume = {{208}},
year = {{2026}},
}
@inbook{60048,
author = {{Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael}},
booktitle = {{Lecture Notes in Computer Science}},
isbn = {{9783031917356}},
issn = {{0302-9743}},
publisher = {{Springer Nature Switzerland}},
title = {{{On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies}}},
doi = {{10.1007/978-3-031-91736-3_19}},
year = {{2025}},
}
@inproceedings{56298,
abstract = {{In the general pattern formation (GPF) problem, a swarm of simple autonomous,
disoriented robots must form a given pattern. The robots' simplicity imply a
strong limitation: When the initial configuration is rotationally symmetric,
only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS
2010]. The only known algorithm to form large patterns with limited visibility
and without memory requires the robots to start in a near-gathering (a swarm of
constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know
any near-gathering algorithm guaranteed to preserve symmetry but most natural
gathering strategies trivially increase symmetries [Castenow et al.; OPODIS
2022].
Thus, we study near-gathering without changing the swarm's rotational
symmetry for disoriented, oblivious robots with limited visibility (the
OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on
the theory of dynamical systems to analyze how a given algorithm affects
symmetry and provide sufficient conditions for symmetry preservation. Until
now, it was unknown whether the considered OBLOT-model allows for any
non-trivial algorithm that always preserves symmetry. Our first result shows
that a variant of Go-to-the-Average always preserves symmetry but may sometimes
lead to multiple, unconnected near-gathering clusters. Our second result is a
symmetry-preserving near-gathering algorithm that works on swarms with a convex
boundary (the outer boundary of the unit disc graph) and without holes (circles
of diameter 1 inside the boundary without any robots).}},
author = {{Gerlach, Raphael and von der Gracht, Sören and Hahn, Christopher and Harbig, Jonas and Kling, Peter}},
booktitle = {{28th International Conference on Principles of Distributed Systems (OPODIS 2024)}},
editor = {{Bonomi, Silvia and Galletta, Letterio and Rivière, Etienne and Schiavoni, Valerio}},
isbn = {{978-3-95977-360-7}},
issn = {{1868-8969}},
keywords = {{Swarm Algorithm, Swarm Robots, Distributed Algorithm, Pattern Formation, Limited Visibility, Oblivious}},
location = {{Lucca, Italy}},
publisher = {{Schloss Dagstuhl -- Leibniz-Zentrum für Informatik}},
title = {{{Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility}}},
doi = {{10.4230/LIPIcs.OPODIS.2024.13}},
volume = {{324}},
year = {{2025}},
}
@unpublished{58953,
abstract = {{In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze their collective Fsync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of R^2. These form a group that is isomorphic to O(2) x S_n, the product group of the orthogonal group and the permutation on n elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear.}},
author = {{Gerlach, Raphael and von der Gracht, Sören}},
booktitle = {{arXiv:2503.07576}},
keywords = {{dynamical systems, coupled systems, distributed computing, robot swarms, autonomous mobile robots, symmetry, equivariant dynamics}},
pages = {{23}},
title = {{{Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems}}},
year = {{2025}},
}
@article{57472,
abstract = {{In this paper we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions toward a weak Pareto optimal solution. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. This work extends results from single objective convex optimization into the multiobjective setting.}},
author = {{Bot, Radu Ioan and Sonntag, Konstantin}},
journal = {{Journal of Mathematical Analysis and Applications}},
keywords = {{Pareto optimization, Lyapunov analysis, gradient-like dynamical systems, inertial dynamics, asymptotic vanishing damping, Tikhonov regularization, strong convergence}},
title = {{{Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization}}},
year = {{2025}},
}
@phdthesis{62750,
abstract = {{Diese Dissertation enthält Beiträge zum Bereich der Mehrzieloptimierung mit einem Fokus auf unbeschränkten Problemen, die auf einem allgemeinen Hilbertraum definiert sind. Für Mehrzieloptimierungsprobleme mit lokal Lipschitz-stetigen Zielfunktionen definieren wir ein multikriterielles Subdifferential, das wir erstmals im Kontext allgemeiner Hilberträume analysieren. Aufbauend auf diesen theoretischen Untersuchungen präsentieren wir ein Abstiegsverfahren, bei welchem in jeder Iteration eine Abstiegsrichtung mittels einer numerischen Approximation des multikriteriellen Subdifferentials bestimmt wird. Im Kontext konvexer, stetig differenzierbarer Zielfunktionen mit Lipschitz-stetigen Gradienten, führen wir eine Familie von dynamischen Gradientensystemen mit Trägheitsterm ein, die bekannte kontinuierliche Systeme aus der skalaren Optimierung verallgemeinern. Wir stellen drei neue Systeme vor: eines mit konstanter Dämpfung, eines mit asymptotisch abnehmender Dämpfung und eines, das zusätzlich eine zeitabhängige Tikhonov-Regularisierung beinhaltet. Aufbauend auf den Untersuchungen der neuen dynamischen Gradientensysteme, entwickeln wir ein beschleunigtes Gradientenverfahren zur Mehrzieloptimierung, das auf einer Diskretisierung des multikriteriellen Gradientensystems mit asymptotisch abnehmender Dämpfung beruht. Das hergeleitete Verfahren bewahrt die günstigen Konvergenzeigenschaften des kontinuierlichen Systems und erreicht eine schnellere Konvergenz als klassische Verfahren.}},
author = {{Sonntag, Konstantin}},
publisher = {{Paderborn University}},
title = {{{First-order methods and gradient dynamical systems for multiobjective optimization}}},
doi = {{10.17619/UNIPB/1-2457}},
year = {{2025}},
}
@article{51208,
abstract = {{AbstractApproximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.}},
author = {{Gebken, Bennet}},
issn = {{0926-6003}},
journal = {{Computational Optimization and Applications}},
keywords = {{Applied Mathematics, Computational Mathematics, Control and Optimization}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A note on the convergence of deterministic gradient sampling in nonsmooth optimization}}},
doi = {{10.1007/s10589-024-00552-0}},
year = {{2024}},
}
@article{46019,
abstract = {{We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, which trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first-order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent.}},
author = {{Sonntag, Konstantin and Peitz, Sebastian}},
journal = {{Journal of Optimization Theory and Applications}},
publisher = {{Springer}},
title = {{{Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems}}},
doi = {{10.1007/s10957-024-02389-3}},
year = {{2024}},
}
@unpublished{51334,
abstract = {{The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.}},
author = {{Sonntag, Konstantin and Gebken, Bennet and Müller, Georg and Peitz, Sebastian and Volkwein, Stefan}},
booktitle = {{arXiv:2402.06376}},
title = {{{A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces}}},
year = {{2024}},
}
@article{52726,
abstract = {{Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.}},
author = {{Bick, Christian and von der Gracht, Sören}},
issn = {{2051-1329}},
journal = {{Journal of Complex Networks}},
keywords = {{Applied Mathematics, Computational Mathematics, Control and Optimization, Management Science and Operations Research, Computer Networks and Communications}},
number = {{2}},
publisher = {{Oxford University Press (OUP)}},
title = {{{Heteroclinic dynamics in network dynamical systems with higher-order interactions}}},
doi = {{10.1093/comnet/cnae009}},
volume = {{12}},
year = {{2024}},
}
@article{32447,
abstract = {{We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas previously laid out in [H. Attouch and G. Garrigos, Multiobjective Optimization: An Inertial Dynamical Approach to Pareto Optima, preprint, arXiv:1506.02823, 2015]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order \(\mathcal{O}(t^{-2})\) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization.}},
author = {{Sonntag, Konstantin and Peitz, Sebastian}},
issn = {{1095-7189}},
journal = {{SIAM Journal on Optimization}},
keywords = {{multiobjective optimization, Pareto optimization, Lyapunov analysis, gradient-likedynamical systems, inertial dynamics, asymptotic vanishing damping, fast convergence}},
number = {{3}},
pages = {{2259 -- 2286}},
publisher = {{Society for Industrial and Applied Mathematics}},
title = {{{Fast Convergence of Inertial Multiobjective Gradient-Like Systems with Asymptotic Vanishing Damping}}},
doi = {{10.1137/23M1588512}},
volume = {{34}},
year = {{2024}},
}
@article{59171,
abstract = {{To model dynamical systems on networks with higher-order (non-pairwise) interactions, we recently introduced a new class of ordinary differential equations (ODEs) on hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady-state branch.}},
author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
issn = {{1364-5021}},
journal = {{Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}},
keywords = {{higher-order interactions, synchrony breaking, network dynamics, coupled cell systems}},
number = {{2301}},
publisher = {{The Royal Society}},
title = {{{Higher-order interactions lead to ‘reluctant’ synchrony breaking}}},
doi = {{10.1098/rspa.2023.0945}},
volume = {{480}},
year = {{2024}},
}
@article{21199,
abstract = {{As in almost every other branch of science, the major advances in data
science and machine learning have also resulted in significant improvements
regarding the modeling and simulation of nonlinear dynamical systems. It is
nowadays possible to make accurate medium to long-term predictions of highly
complex systems such as the weather, the dynamics within a nuclear fusion
reactor, of disease models or the stock market in a very efficient manner. In
many cases, predictive methods are advertised to ultimately be useful for
control, as the control of high-dimensional nonlinear systems is an engineering
grand challenge with huge potential in areas such as clean and efficient energy
production, or the development of advanced medical devices. However, the
question of how to use a predictive model for control is often left unanswered
due to the associated challenges, namely a significantly higher system
complexity, the requirement of much larger data sets and an increased and often
problem-specific modeling effort. To solve these issues, we present a universal
framework (which we call QuaSiModO:
Quantization-Simulation-Modeling-Optimization) to transform arbitrary
predictive models into control systems and use them for feedback control. The
advantages of our approach are a linear increase in data requirements with
respect to the control dimension, performance guarantees that rely exclusively
on the accuracy of the predictive model, and only little prior knowledge
requirements in control theory to solve complex control problems. In particular
the latter point is of key importance to enable a large number of researchers
and practitioners to exploit the ever increasing capabilities of predictive
models for control in a straight-forward and systematic fashion.}},
author = {{Peitz, Sebastian and Bieker, Katharina}},
journal = {{Automatica}},
publisher = {{Elsevier}},
title = {{{On the Universal Transformation of Data-Driven Models to Control Systems}}},
doi = {{10.1016/j.automatica.2022.110840}},
volume = {{149}},
year = {{2023}},
}
@article{49326,
abstract = {{Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.}},
author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
issn = {{0036-1399}},
journal = {{SIAM Journal on Applied Mathematics}},
keywords = {{Applied Mathematics}},
number = {{6}},
pages = {{2329--2353}},
publisher = {{Society for Industrial & Applied Mathematics (SIAM)}},
title = {{{Hypernetworks: Cluster Synchronization Is a Higher-Order Effect}}},
doi = {{10.1137/23m1561075}},
volume = {{83}},
year = {{2023}},
}
@unpublished{46578,
abstract = {{Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".}},
author = {{Bernreuther, Marco and Dellnitz, Michael and Gebken, Bennet and Müller, Georg and Peitz, Sebastian and Sonntag, Konstantin and Volkwein, Stefan}},
booktitle = {{arXiv:2308.01113}},
title = {{{Multiobjective Optimization of Non-Smooth PDE-Constrained Problems}}},
year = {{2023}},
}
@article{27426,
abstract = {{Regularization is used in many different areas of optimization when solutions
are sought which not only minimize a given function, but also possess a certain
degree of regularity. Popular applications are image denoising, sparse
regression and machine learning. Since the choice of the regularization
parameter is crucial but often difficult, path-following methods are used to
approximate the entire regularization path, i.e., the set of all possible
solutions for all regularization parameters. Due to their nature, the
development of these methods requires structural results about the
regularization path. The goal of this article is to derive these results for
the case of a smooth objective function which is penalized by a piecewise
differentiable regularization term. We do this by treating regularization as a
multiobjective optimization problem. Our results suggest that even in this
general case, the regularization path is piecewise smooth. Moreover, our theory
allows for a classification of the nonsmooth features that occur in between
smooth parts. This is demonstrated in two applications, namely support-vector
machines and exact penalty methods.}},
author = {{Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}},
journal = {{Journal of Global Optimization}},
number = {{3}},
pages = {{709--741}},
title = {{{On the structure of regularization paths for piecewise differentiable regularization terms}}},
doi = {{10.1007/s10898-022-01223-2}},
volume = {{85}},
year = {{2023}},
}
@unpublished{45498,
abstract = {{We present a novel method for high-order phase reduction in networks of
weakly coupled oscillators and, more generally, perturbations of reducible
normally hyperbolic (quasi-)periodic tori. Our method works by computing an
asymptotic expansion for an embedding of the perturbed invariant torus, as well
as for the reduced phase dynamics in local coordinates. Both can be determined
to arbitrary degrees of accuracy, and we show that the phase dynamics may
directly be obtained in normal form. We apply the method to predict remote
synchronisation in a chain of coupled Stuart-Landau oscillators.}},
author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
booktitle = {{arXiv:2306.03320}},
pages = {{29}},
title = {{{A parametrisation method for high-order phase reduction in coupled oscillator networks}}},
year = {{2023}},
}
@article{23428,
abstract = {{The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d.
samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the
best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.}},
author = {{Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}},
journal = {{Journal of Nonlinear Science}},
title = {{{Finite-data error bounds for Koopman-based prediction and control}}},
doi = {{10.1007/s00332-022-09862-1}},
volume = {{33}},
year = {{2023}},
}
@article{21600,
abstract = {{Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.}},
author = {{Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}},
journal = {{SIAM Journal on Scientific Computing}},
number = {{2}},
pages = {{A579--A595}},
title = {{{Efficient time stepping for numerical integration using reinforcement learning}}},
doi = {{10.1137/21M1412682}},
volume = {{45}},
year = {{2023}},
}