Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves - Offen, Christian; Ober-Blöbaum, Sina (Geometric Science of Information)
The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.
Keywords: System identification; discrete Lagrangians; travelling waves

Variational Learning of Euler–Lagrange Dynamics from Data - Ober-Blöbaum, Sina; Offen, Christian (Journal of Computational and Applied Mathematics)
The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.
Keywords: Lagrangian learning; variational backward error analysis; modified Lagrangian; variational integrators; physics informed learning

Backward error analysis for conjugate symplectic methods - McLachlan, Robert; Offen, Christian (Journal of Geometric Mechanics)
The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
Keywords: variational integrators; backward error analysis; Euler--Lagrange equations; multistep methods; conjugate symplectic methods

Hamiltonian Neural Networks with Automatic Symmetry Detection - Dierkes, Eva; Offen, Christian; Ober-Blöbaum, Sina; Flaßkamp, Kathrin (Chaos)
Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.

Testinstrumente und Testdatenanalyse zur Verarbeitung von Unsicherheiten in Logikblöcken hochintegrierter Schaltungen - Sprenger, Alexander
Ever increasing demands on the performance of microchips are leading to ever more complex semiconductor technologies with ever shrinking feature sizes. Complex applications with high demands on safety and reliability, such as autonomous driving, are simultaneously driving the requirements for test and diagnosis of VLSI circuits. Throughout the life cycle of a microchip, uncertainties occur that affect its timing behavior. For example, weak circuit structures, aging effects, or process variations can lead to a change in the timing behavior of the circuit. While these uncertainties do not necessarily lead to a change of the functional behavior, they can lead to a reliability problem. With modular and hybrid compaction two test instruments are presented in this work that can be used for X-tolerant test response compaction in the built-in Faster-than-At-Speed Test (FAST) which is used to detect uncertainties in VLSI circuits. One challenge for test response compaction during FAST is the high and varying X-rate at the outputs of the circuit under test. By dividing the circuit outputs into test groups and separately compacting these test groups using stochastic compactors, the modular compaction is able to handle these high and varying X-rates. To deal with uncertainties on logic interconnects, a method for distinguishing crosstalk and process variation is presented. In current semiconductor technologies, the number of parasitic coupling capacitances between logic interconnects is growing. These coupling capacitances can lead to crosstalk, which causes increased current flow in the logic interconnects, which in turn can lead to increased electromigration. In the presented method, delay maps describing the timing behavior of the circuit outputs at different operating points are used to train artificial neural networks which classify the tested circuits into fault-free and faulty.
Keywords: Testantwortkompaktierung; Prozessvariation; Silicon Lifecycle Management

Partial observations, coarse graining and equivariance in Koopman operator theory for large-scale dynamical systems - Peitz, Sebastian; Harder, Hans; Nüske, Feliks; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl (arXiv:2307.15325)
The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems, the main reason being the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. Until now, the situation where for large-scale systems, we (i) only have access to partial observations (i.e., measurements, as is very common for experimental data) or (ii) deliberately perform coarse graining (for efficiency reasons) has not been treated to its full extent. In this paper, we address the pitfall associated with this situation, that the classical EDMD algorithm does not automatically provide a Koopman operator approximation for the underlying system if we do not carefully select the number of observables. Moreover, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to massively increase the model efficiency. We also briefly draw a connection to domain decomposition techniques for partial differential equations and present numerical evidence using the Kuramoto--Sivashinsky equation.

Revisiting the initial reaction rates for TMS combustion and a new evidence for metastable silica nanoparticles in the gas-phase synthesis - Karakaya, Yasin; Somnitz, H.; Hermsen, A.; Gonchikzhapov, Munko; Kasper, Tina (Applications in Energy and Combustion Science)
Keywords: Metastable particles; Nanomaterials; Quartz-crystal-microbalance; Molecular-beam mass spectrometry; Tetramethylsilane; Reaction kinetics

Finite-data error bounds for Koopman-based prediction and control - Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl (Journal of Nonlinear Science)
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.

Multi-Objective Optimization for Sparse Deep Neural Network Training - Hotegni, Sedjro Salomon; Peitz, Sebastian; Berkemeier, Manuel Bastian (arXiv:2308.12243)
Different conflicting optimization criteria arise naturally in various Deep Learning scenarios. These can address different main tasks (i.e., in the setting of Multi-Task Learning), but also main and secondary tasks such as loss minimization versus sparsity. The usual approach is a simple weighting of the criteria, which formally only works in the convex setting. In this paper, we present a Multi-Objective Optimization algorithm using a modified Weighted Chebyshev scalarization for training Deep Neural Networks (DNNs) with respect to several tasks. By employing this scalarization technique, the algorithm can identify all optimal solutions of the original problem while reducing its complexity to a sequence of single-objective problems. The simplified problems are then solved using an Augmented Lagrangian method, enabling the use of popular optimization techniques such as Adam and Stochastic Gradient Descent, while efficaciously handling constraints. Our work aims to address the (economical and also ecological) sustainability issue of DNN models, with a particular focus on Deep Multi-Task models, which are typically designed with a very large number of weights to perform equally well on multiple tasks. Through experiments conducted on two Machine Learning datasets, we demonstrate the possibility of adaptively sparsifying the model during training without significantly impacting its performance, if we are willing to apply task-specific adaptations to the network weights. Code is available at https://github.com/salomonhotegni/MDMTN.

Thermal and chemical structure of ethanol and 2-ethylhexanoic acid/ethanol SpraySyn flames - Gonchikzhapov, Munko; Kasper, Tina (Applications in Energy and Combustion Science)
Keywords: Nanoparticle synthesis; Flame spray pyrolysis; SpraySyn burner; Flame structure; Species distribution; Temperature distribution

ElectricGrid.jl - A Julia-based modeling and simulationtool for power electronics-driven electric energy grids - Wallscheid, Oliver; Peitz, Sebastian; Stenner, Jan; Weber, Daniel; Boshoff, Septimus; Meyer, Marvin; Chidananda, Vikas; Schweins, Oliver (Journal of Open Source Software)
Keywords: General Earth and Planetary Sciences; General Environmental Science

Paradigmenwechsel vom klassischen zum datengetriebenen Problemlösen im Informatikunterricht - Höper, Lukas; Schulte, Carsten (MNU journal)

On the impossibility of surviving (iterated) deletion of weakly dominated strategies in rational MPC - Blömer, Johannes; Bobolz, Jan; Bröcher, Henrik

A Generic Construction of an Anonymous Reputation System and Instantiations from Lattices - Blömer, Johannes; Bobolz, Jan; Porzenheim, Laurens Alexander

Influence of the filler–matrix adhesion and the effects of conditioning on tensile properties of laser-sintered parts built with polyamide–glass bead dry blends - Kletetzka, Ivo; Kosanke, Maren; Meinderink, Dennis; Neßlinger, Vanessa; Grundmeier, Guido; Schmid, Hans-Joachim (Progress in Additive Manufacturing)
In this work, the influence of the filler–matrix adhesion on the tensile properties of laser-sintered parts built with Polyamide 613 filled with glass beads was investigated. For this purpose, dry blends of glass beads with and without organosilane coupling agents and polyamide powder were prepared and processed into tensile specimens on an EOS P396 laser sintering system. The samples were tested both in the dry state and after an accelerated conditioning in a climate chamber. Furthermore, finite element method (FEM) simulations were performed to model the extreme cases of optimum adhesion and no adhesion. By correlating the tensile tests with the simulation results and by analyzing the fracture surfaces, it was shown that the filler–matrix adhesion is sufficient in the dry state but is strongly degraded by conditioning. Even the presence of various organosilane thin films could not prevent a strong deterioration of the filler–matrix adhesion and the associated deterioration of the mechanical properties. Since a comparison with an injection-molded sample of the same polymer filler combination shows identical behavior after conditioning, it is assumed that this problem is not limited to additively manufactured parts.
Keywords: Industrial and Manufacturing Engineering

Influence of the filler matrix adhesion and the effects of conditioning on tensile properties of laser sintered parts built with polyamide glass bead dry blends - Kletetzka, Ivo; Kosanke, Maren; Meinderink, Dennis; Neßlinger, Vanessa; Grundmeier, Guido; Schmid, Hans-Joachim
Abstract In this work, the influence of the filler matrix adhesion on the tensile properties of laser-sintered parts built with Polyamide 613 filled with glass beads was investigated. For this purpose, dry blends of glass beads with and without organosilane coupling agents and polyamide powder were prepared and processed into tensile specimens on an EOS P396. The samples were tested both in the dry state and after an accelerated conditioning in a climate chamber. Furthermore, FEM simulations were performed to model the limiting cases of optimum adhesion and no adhesion. By correlating the tensile tests with the simulation results and by analyzing the fracture surfaces, it was shown that the filler matrix adhesion is sufficient in the dry state but is strongly degraded by conditioning. Even the presence of various organosilane thin films could not prevent a strong deterioration of the filler-matrix adhesion and the associated deterioration of the mechanical properties. Since a comparison with an injection molded sample of the same polymer filler combination shows identical behavior after conditioning, it is assumed that this problem is not limited to additively manufactured parts.

Influence of a prolonged shelf time on PA12 laser sintering powder and resulting part properties - Klippstein, Sven Helge; Kletetzka, Ivo; Sural, Ilknur; Schmid, Hans-Joachim (The International Journal of Advanced Manufacturing Technology)
In the laser sintering technology, the semi-crystalline polymer material is exposed to elevated temperatures during processing, which leads to serious material ageing for most materials. This has already been investigated intensively by various authors. However, the ageing of the material at ambient temperatures during shelf life has not been the focus so far. The need to analyse the shelf life can be derived from an ecological and economic point of view. This work is focusing on the shelf life of PA2200 (PA12). To reduce the potential influences of powder production fluctuations, two different powder batches stored for 5.5 years and 6.5 years are investigated and compared to a reference powder produced 0.5 years before these investigations. Multiple powder analyses and part characterisations have been performed. A significant yellowing and molecular chain length reduction can be derived from the measurement results. Whereas the influence on mechanical part performance was minor, the parts built with the stored powders are more yellowish. As it is most likely that this is due to the consumption of polyamide stabilisers, it can be assumed that these parts will be subject to significantly faster ageing. Therefore, it is still not recommended to use the stored powders for critical parts or light intense and humid environments.
Keywords: Selective laser sintering; Shelf life; Polyamide 12; powder; PA2200; material ageing

Editorial - Schönhärl, Korinna; Schotters, Frederike; Thiemeyer, Guido (Werkstatt Geschichte)
»Über Geld spricht man nicht.« Diese Benimmregel erweist sich beim Blick in die Ver- gangenheit als Illusion. Keineswegs war Geld grundsätzlich ein Tabuthema zwischen- menschlicher Kommunikation. Ganz im Gegenteil: Mit dem Reden über Geld wird dieses mit Bedeutung aufgeladen. Die Autor*innen dieses Themenheftes untersuch en Gelddis- kurse und die Zuschreibungen von Bedeutung an Geld in der internationalen Geschichte.
Keywords: Financial History; Discourse Analysis; History of Money

Transferability of a discrepancy model for the dynamics of electromagnetic oscillating circuits - Wohlleben, Meike Claudia; Muth, Lars; Peitz, Sebastian; Sextro, Walter (Proceedings in Applied Mathematics and Mechanics)
Modelling of dynamic systems plays an important role in many engineering disciplines. Two different approaches are physical modelling and data‐driven modelling, both of which have their respective advantages and disadvantages. By combining these two approaches, hybrid models can be created in which the respective disadvantages are mitigated, with discrepancy models being a particular subclass. Here, the basic system behaviour is described physically, that is, in the form of differential equations. Inaccuracies resulting from insufficient modelling or numerics lead to a discrepancy between the measurements and the model, which can be compensated by a data‐driven error correction term. Since discrepancy methods still require a large amount of measurement data, this paper investigates the extent to which a single discrepancy model can be trained for a physical model with additional parameter dependencies without the need for retraining. As an example, a damped electromagnetic oscillating circuit is used. The physical model is realised by a differential equation describing the electric current, considering only inductance and capacitance; dissipation due to resistance is neglected. This creates a discrepancy between measurement and model, which is corrected by a data‐driven model. In the experiments, the inductance and the capacity are varied. It is found that the same data‐driven model can only be used if additional parametric dependencies in the data‐driven term are considered as well.
Keywords: Electrical and Electronic Engineering; Atomic and Molecular Physics; and Optics

Expert-Informed Hierarchical Diagnostics of Multiple Fault Modes of a Spacecraft Propulsion System - Aimiyekagbon, Osarenren Kennedy; Löwen, Alexander; Bender, Amelie; Muth, Lars; Sextro, Walter (Proceedings of the Asia Pacific Conference of the PHM Society 2023)
This paper presents a comprehensive study on diagnosing a spacecraft propulsion system utilizing data provided by the Prognostics and Health Management (PHM) society, specifically obtained as part of the Asia-Pacific PHM conference’s data challenge 2023. The objective of the challenge is to identify and diagnose known faults as well as unknown anomalies in the spacecraft’s propulsion system, which is critical for ensuring the spacecraft’s proper functionality and safety. To address this challenge, the proposed method follows a systematic approach of feature extraction, feature selection, and model development. The models employed in this study are kMeans clustering and decision trees combined to ensembles, enriched with expert knowledge. With the method presented, our team was capable of reaching high accuracy in identifying anomalies as well as diagnosing faults, resulting in attaining the seventh place with a score of 93.08 %.
Keywords: PHM; Fault Diagnostics; Multiple Fault Modes; Expert-Informed Diagnostics; Anomaly Detection