@unpublished{55159, abstract = {{We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schrödinger equation. The article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.}}, author = {{Offen, Christian}}, keywords = {{System identification, inverse problem of variational calculus, Gaussian process, Lagrangian learning, physics informed machine learning, geometry aware learning}}, pages = {{28}}, title = {{{Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification}}}, year = {{2024}}, } @article{45972, author = {{Kovács, Balázs}}, journal = {{SIAM Journal on Scientific Computing}}, number = {{2}}, pages = {{A645----A669}}, title = {{{Numerical surgery for mean curvature flow of surfaces}}}, doi = {{10.1137/22M1531919}}, volume = {{46}}, year = {{2024}}, } @article{56497, author = {{Cappello, Chiara and Naserasr, Reza and Steffen, Eckhard and Wang, Zhouningxin}}, issn = {{0012-365X}}, journal = {{Discrete Mathematics}}, number = {{1}}, publisher = {{Elsevier BV}}, title = {{{Critically 3-frustrated signed graphs}}}, doi = {{10.1016/j.disc.2024.114258}}, volume = {{348}}, year = {{2024}}, } @inproceedings{59791, author = {{Maslovskaya, Sofya and Ober-Blöbaum, Sina}}, booktitle = {{IFAC-PapersOnLine}}, issn = {{2405-8963}}, number = {{17}}, pages = {{85--90}}, publisher = {{Elsevier BV}}, title = {{{Symplectic Methods in Deep Learning}}}, doi = {{10.1016/j.ifacol.2024.10.118}}, volume = {{58}}, year = {{2024}}, } @unpublished{59801, author = {{Jean, Frédéric and Maslovskaya, Sofya}}, title = {{{Inverse optimal control problem in the non autonomous linear-quadratic case}}}, year = {{2024}}, } @unpublished{55078, abstract = {{This paper develops and discusses a residual-based a posteriori error estimate and a space--time adaptive algorithm for solving parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. A space--time adaptive algorithm is proposed using the derived error indicator. Numerical experiments illustrate and complement the theory.}}, author = {{Kovács, Balázs and Lantelme, Michael Frederik Raúl}}, booktitle = {{arXiv:2407.02101}}, title = {{{A posteriori error estimates for parabolic partial differential equations on stationary surfaces}}}, year = {{2024}}, } @unpublished{56429, abstract = {{It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the hyperoctahedral group for $G=C_2$. We give structural characterisations of transitive subsets using the character theory of $G \wr S_n$ and interpret such subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In particular, we prove a generalisation of the Livingstone-Wagner theorem and give explicit constructions of transitive sets. Moreover, we establish connections to orthogonal polynomials, namely the Charlier polynomials, and use them to study codes and designs in $C_r \wr S_n$. Many of our results extend results about the symmetric group $S_n$.}}, author = {{Klawuhn, Lukas-André Dominik and Schmidt, Kai-Uwe}}, booktitle = {{arXiv:2409.20495}}, pages = {{38}}, title = {{{Transitivity in wreath products with symmetric groups}}}, 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}}, } @article{49372, author = {{Klüners, Jürgen and Wang, Jiuya}}, issn = {{2730-9657}}, journal = {{La Matematica}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Idélic Approach in Enumerating Heisenberg Extensions}}}, doi = {{10.1007/s44007-023-00067-w}}, year = {{2023}}, } @inproceedings{34135, abstract = {{By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise.}}, author = {{Lishkova, Yana and Scherer, Paul and Ridderbusch, Steffen and Jamnik, Mateja and Liò, Pietro and Ober-Blöbaum, Sina and Offen, Christian}}, booktitle = {{IFAC-PapersOnLine}}, location = {{ Yokohama, Japan}}, number = {{2}}, pages = {{3203--3210}}, publisher = {{Elsevier}}, title = {{{Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery}}}, doi = {{10.1016/j.ifacol.2023.10.1457}}, volume = {{56}}, year = {{2023}}, } @article{50298, abstract = {{A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.}}, author = {{Schmidt, Kai-Uwe and Weiß, Charlene}}, journal = {{Combinatorial Theory}}, number = {{1}}, title = {{{Packings and Steiner systems in polar spaces}}}, doi = {{10.5070/c63160424}}, volume = {{3}}, year = {{2023}}, } @article{50297, abstract = {{We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled.}}, author = {{Schmidt, Kai‐Uwe and Weiß, Charlene}}, journal = {{Journal of Combinatorial Designs}}, number = {{9}}, pages = {{422--431}}, publisher = {{Wiley}}, title = {{{Existence of small ordered orthogonal arrays}}}, doi = {{10.1002/jcd.21903}}, volume = {{31}}, year = {{2023}}, } @phdthesis{50300, abstract = {{Digital communications relies heavily on the usage of different types of codes. Prominent codes nowadays are rank-metric codes and subspace codes - the q-analogs of binary codes and binary codes with constant weight. All these codes can be viewed as subsets of classical association schemes. A central coding-theoretic problem is to derive upper bounds for the size of codes. This thesis investigates Delsartes powerful linear program whose optimum is precisely such a bound for codes in association schemes. The linear programs for binary codes and binary constant-weight codes have been extensively studied since the 1970s, but their optimum is still unknown. We determine in a unified way the optimum of the linear program in several ordinary q-analogs as well as in their affine counterparts. In particular, bounds and constructions for codes in polar spaces are established, where the bounds are sharp up to a constant factor in many cases. Moreover, based on these results, an almost complete classification of Steiner systems in polar spaces is provided by showing that they could only exist in some corner cases.}}, author = {{Weiß, Charlene}}, title = {{{Linear programming bounds in classical association schemes}}}, doi = {{10.17619/UNIPB/1-1672}}, year = {{2023}}, } @article{51351, author = {{Steffen, Eckhard and Wolf, Isaak Hieronymus}}, issn = {{0166-218X}}, journal = {{Discrete Applied Mathematics}}, keywords = {{Applied Mathematics, Discrete Mathematics and Combinatorics}}, pages = {{185--189}}, publisher = {{Elsevier BV}}, title = {{{Bounds for the chromatic index of signed multigraphs}}}, doi = {{10.1016/j.dam.2023.05.008}}, volume = {{337}}, year = {{2023}}, } @inbook{45190, author = {{Cappello, Chiara and Steffen, Eckhard}}, booktitle = {{The Digital Twin of Humans}}, isbn = {{9783031261039}}, pages = {{93----110}}, publisher = {{Springer International Publishing}}, title = {{{Graph-Theoretical Models for the Analysis and Design of Socio-Technical Networks}}}, doi = {{10.1007/978-3-031-26104-6_5}}, year = {{2023}}, } @article{51357, author = {{Steffen, Eckhard and Wolf, Isaak Hieronymus}}, issn = {{0012-365X}}, journal = {{Discrete Mathematics}}, keywords = {{Discrete Mathematics and Combinatorics, Theoretical Computer Science}}, publisher = {{Elsevier BV}}, title = {{{Rotation r-graphs}}}, doi = {{10.1016/j.disc.2023.113457}}, 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}}, } @book{45191, editor = {{Gräßler, Iris and Maier, Günter W. and Steffen, Eckhard and Roesmann, Daniel}}, isbn = {{9783031261039}}, publisher = {{Springer International Publishing}}, title = {{{The Digital Twin of Humans}}}, doi = {{10.1007/978-3-031-26104-6}}, year = {{2023}}, }