Structure-preserving integration

Simulations of dynamical systems are intended to reproduce the dynamic behavior in a realistic way. Using structure-preserving integration schemes for the simulation of mechanical systems, certain properties of the real system are conserved in the numerical solution. Examples are the conservation of energy or momentum induced by symmetries in the system (e.g., conservation of the angular momentum in case of rotational symmetry). A special class of structure-preserving integrators are variational integrators that are derived based on discrete variational principles. As it is well-known, the fulfilment of a variational principle leads to the Euler-Lagrange equations of motion describing the dynamics of mechanical or electrical systems. Variational integrators are constructed by directly discretizing the variational principle rather than the Euler-Lagrange equations. The resulting time-stepping schemes are symplectic by construction which leads to excellent long time energy behavior. Furthermore, the fulfilment of the discrete Noether theorem guarantees preservation of momentum maps induced by symmetries.

In our research group we work on higher and mixed order constructions and the analysis of variational integrators as well as their extension to various system classes. The construction of numerical methods of higher order yields so called Galerkin variational integrators whose numerical properties are analyzed in depth by means of particularly tailored variational methods. System classes we consider are, for example, dynamical systems on different time scales, fractional Lagrangian systems, electric circuits and flexible beam dynamics.

Read more about our work on

• Higher and mixed order Galerkin variational integrators
• Variational multirate integration for dynamics on different time scales
• Variational integrators for fractional Lagrangians
• Variational Lie-group integrators for flexible beams
• Variational integrators for electric circuits

Optimal Control

Optimal control aims to prescribe the motion of a dynamical system in such a way that a certain optimality criterion is achieved. The research focus lies in the development of efficient numerical schemes for the solution of optimal control problems that are based on structure-preserving integration. In particular, optimal control methods are designed for the treatment of multi-body systems as well as for complex systems with certain substructures for which hierarchical approaches are developed. Further aspects of research interest are the development of numerical methods using inherent properties of the dynamical system such as symmetries or invariant objects, multi-objective optimization and model predictive approaches for optimal control problems and the optimal control of hybrid systems. Besides the investigation of theoretical aspects regarding accuracy and convergence of the numerical schemes, their performance is validated by means of problems from different fields of applications, e.g., mechatronic systems, biomechanics and astrodynamics.

• Discrete mechanics and optimal control (DMOC)
• Exploiting symmetry in optimal control
• Primitives and multi-objective model predictive control
• Turnpikes in optimal control
• Safety-Critical control of High Dimensional Systems
• Optimal control of multirate dynamical systems
• Optimal control of hybrid systems and mixed-integer problems
• Optimal control approach to deep learning

Learning of Hamiltonians, Lagrangians, and Symmetries

Economical or biological models, and models of physical systems are described by differential equations. In practice, these governing equations are often either unknown or only partial information is available. Using machine learning, missing knowledge about models can be learned from observational data. However, it appears natural to incorporate the prior knowledge that we have as this will make a learned model more reliable and reduce data requirements. In our current research, we develop new methods to incorporate prior knowledge about the validity of fundamental physical principles into machine learned models of dynamical systems.

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• Hamilton's Principle and Machine Learning

Photonic Quantum Computing/PhoQC (MKW)

• PhoQC

Symplectic discretizations for optimal control problems in mechanics, DFG, 2023-2025

• Selfoptimizing systems in mechanical engineering: model-oriented self-optimization, (DFG)-research project SFB 614, A1, 2009 - 2013

• Spitzencluster it's OWL Intelligente Technische Systeme OstWestfalenLippe, Querschnittsprojekt Selbstoptimierung, BMBF, 2012 - 2015

• Fractional Variational Integration and Optimal Control EPSRC-project Fractional Variational Integration and Optimal Control, EPSRC, UK, 2017-2021

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