Carsten Hartmann

Burkhard Schmidt, Carsten Hartmann

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schrödinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm.

Simon Becker, Carsten Hartmann

Along the ideas of Curtain and Glover, we extend the balanced truncation method for infinite-dimensional linear systems to bilinear and stochastic systems. Specifically , we apply Hilbert space techniques used in many-body quantum mechanics to establish error bounds for the truncated system and prove convergence results. The functional analytic setting allows us to obtain mixed Hardy space error bounds for both finite-and infinite-dimensional systems, and it is then applied to the model reduction of stochastic evolution equations driven by Wiener noise.

Ashkan Mansouri Yarahmadi, Michael Breuß, Carsten Hartmann

In context of laser powder bed fusion (L-PBF), it is known that the properties of the final fabricated product highly depend on the temperature distribution and its gradient over the manufacturing plate. In this paper, we propose a novel means to predict the temperature gradient distributions during the printing process by making use of neural networks. This is realized by employing heat maps produced by an optimized printing protocol simulation and used for training a specifically tailored recurrent neural network in terms of a long short-term memory architecture. The aim of this is to avoid extreme and inhomogeneous temperature distribution that may occur across the plate in the course of the printing process. In order to train the neural network, we adopt a well-engineered simulation and unsupervised learning framework. To maintain a minimized average thermal gradient across the plate, a cost function is introduced as the core criteria, which is inspired and optimized by considering the well-known traveling salesman problem (TSP). As time evolves the unsupervised printing process governed by TSP produces a history of temperature heat maps that maintain minimized average thermal gradient. All in one, we propose an intelligent printing tool that provides control over the substantial printing process components for L-PBF, i.e.\ optimal nozzle trajectory deployment as well as online temperature prediction for controlling printing quality.

Carsten Hartmann, Lorenz Richter

The recent advances in machine learning in various fields of applications can be largely attributed to the rise of deep learning (DL) methods and architectures. Despite being a key technology behind autonomous cars, image processing, speech recognition, etc., a notorious problem remains the lack of theoretical understanding of DL and related interpretability and (adversarial) robustness issues. Understanding the specifics of DL, as compared to, say, other forms of nonlinear regression methods or statistical learning, is interesting from a mathematical perspective, but at the same time it is of crucial importance in practice: treating neural networks as mere black boxes might be sufficient in certain cases, but many applications require waterproof performance guarantees and a deeper understanding of what could go wrong and why it could go wrong. It is probably fair to say that, despite being mathematically well founded as a method to approximate complicated functions, DL is mostly still more like modern alchemy that is firmly in the hands of engineers and computer scientists. Nevertheless, it is evident that certain specifics of DL that could explain its success in applications demands systematic mathematical approaches. In this work, we review robustness issues of DL and particularly bridge concerns and attempts from approximation theory to statistical learning theory. Further, we review Bayesian Deep Learning as a means for uncertainty quantification and rigorous explainability.

Carsten Hartmann, Christof Schütte

Rare event simulation and estimation for systems in equilibrium are among the most challenging topics in molecular dynamics. As was shown by Jarzynski and others, nonequilibrium forcing can theoretically be used to obtain equilibrium rare event statistics. The advantage seems to be that the external force can speed up the sampling of the rare events by biasing the equilibrium distribution towards a distribution under which the rare events is no longer rare. Yet algorithmic methods based on Jarzynski's and related results often fail to be efficient because they are based on sampling in path space. We present a new method that replaces the path sampling problem by minimization of a cross-entropy-like functional which boils down to finding the optimal nonequilibrium forcing. We show how to solve the related optimization problem in an efficient way by using an iterative strategy based on milestoning.

Vahid Nateghi, Lara Neureither, Selma Moqvist et al.

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Oliver Bensch, Leonie Bensch, Tommy Nilsson et al.

As humanity prepares for new missions to the Moon and Mars, astronauts will need to operate with greater autonomy, given the communication delays that make real-time support from Earth difficult. For instance, messages between Mars and Earth can take up to 24 minutes, making quick responses impossible. This limitation poses a challenge for astronauts who must rely on in-situ tools to access the large volume of data from spacecraft sensors, rovers, and satellites, data that is often fragmented and difficult to use. To bridge this gap, systems like the Mars Exploration Telemetry-Driven Information System (METIS) are being developed. METIS is an AI assistant designed to handle routine tasks, monitor spacecraft systems, and detect anomalies, all while reducing the reliance on mission control. Current Generative Pretrained Transformer (GPT) Models, while powerful, struggle in safety-critical environments. They can generate plausible but incorrect responses, a phenomenon known as "hallucination," which could endanger astronauts. To overcome these limitations, this paper proposes enhancing systems like METIS by integrating GPTs, Retrieval-Augmented Generation (RAG), Knowledge Graphs (KGs), and Augmented Reality (AR). The idea is to allow astronauts to interact with their data more intuitively, using natural language queries and visualizing real-time information through AR. KGs will be used to easily access live telemetry and multimodal data, ensuring that astronauts have the right information at the right time. By combining AI, KGs, and AR, this new system will empower astronauts to work more autonomously, safely, and efficiently during future space missions.

Rania Rayyes, Daniel Kubus, Carsten Hartmann et al.

Online Goal Babbling and Direction Sampling are recently proposed methods for direct learning of inverse kinematics mappings from scratch even in high-dimensional sensorimotor spaces following the paradigm of "learning while behaving". To learn inverse statics mappings - primarily for gravity compensation - from scratch and without using any closed-loop controller, we modify and enhance the Online Goal Babbling and Direction Sampling schemes. Moreover, we exploit symmetries in the inverse statics mappings to drastically reduce the number of samples required for learning inverse statics models. Results for a 2R planar robot, a 3R simplified human arm, and a 4R humanoid robot arm clearly demonstrate that their inverse statics mappings can be learned successfully with our modified online Goal Babbling scheme. Furthermore, we show that the number of samples required for the 2R and 3R arms can be reduced by a factor of at least 8 and 16 resp. -depending on the number of discovered symmetries.

Peter Benner, Tobias Breiten, Carsten Hartmann et al.

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.