arXiv subjects

Michael Kraus

Publications and source records attributed to Michael Kraus.

At least 19 recordsLinked to original sources

arXiv2025

Neural non-canonical Hamiltonian dynamics for long-time simulations

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

cs.LG↗

arXiv2024

Energy-based, geometric, and compositional formulation of fluid and plasma models

Fluid dynamics plays a crucial role in various multiphysics applications, including energy systems, electronics cooling, and biomedical engineering. Developing models for complex coupled systems can be challenging and time-consuming. In particular, ensuring the consistent integration of models from diverse physical domains requires meticulous attention. Considering the example of (electro-)magneto hydrodynamics (on a fixed spatial domain and with linear polarization and magnetization), this article demonstrates how relatively complex models can be composed from simpler parts by means of a formal language for multiphysics modeling. The Exergetic Port-Hamiltonian Systems (EPHS) modeling language features a simple graphical syntax for expressing the energy-based interconnection of subsystems. This reduces cognitive load and facilitates communication, especially in multidisciplinary environments. As the example demonstrates, existing models can be easily integrated as subsystems of new models. Specifically, an ideal fluid model is used as a subsystem of a Navier-Stokes-Fourier fluid model, which in turn is reused as a subsystem of an (electro-)magneto hydrodynamics model. The energy-based, compositional approach simplifies understanding complex models, and it makes it easy to encapsulate, reuse, and replace (parts of) models. Moreover, structural properties of EPHS models guarantee fundamental properties of thermodynamic systems, such as conservation of energy, non-negative entropy production, and Onsager reciprocal relations.

cs.CE↗

arXiv2024

Structure-preserving particle methods for the Landau collision operator using the metriplectic framework

We present a novel family of particle discretisation methods for the nonlinear Landau collision operator. We exploit the metriplectic structure underlying the Vlasov-Maxwell-Landau system in order to obtain disretisation schemes that automatically preserve mass, momentum, and energy, warrant monotonic dissipation of entropy, and are thus guaranteed to respect the laws of thermodynamics. In contrast to recent works that used radial basis functions and similar methods for regularisation, here we use an auxiliary spline or finite element representation of the distribution function to this end. Discrete gradient methods are employed to guarantee the aforementioned properties in the time discrete domain as well.

physics.plasm-ph↗

arXiv2023

Volume-Preserving Transformers for Learning Time Series Data with Structure

Two of the many trends in neural network research of the past few years have been (i) the learning of dynamical systems, especially with recurrent neural networks such as long short-term memory networks (LSTMs) and (ii) the introduction of transformer neural networks for natural language processing (NLP) tasks. While some work has been performed on the intersection of these two trends, those efforts were largely limited to using the vanilla transformer directly without adjusting its architecture for the setting of a physical system. In this work we develop a transformer-inspired neural network and use it to learn a dynamical system. We (for the first time) change the activation function of the attention layer to imbue the transformer with structure-preserving properties to improve long-term stability. This is shown to be of great advantage when applying the neural network to learning the trajectory of a rigid body.

math.NA↗

arXiv2023

Symplectic Autoencoders for Model Reduction of Hamiltonian Systems

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.

cs.LG↗

arXiv2023

A structure-preserving particle discretisation for the Lenard-Bernstein collision operator

Collisions are an important dissipation mechanism in plasmas. In one-dimensional modelling, a commonly used collision operator is the Lenard-Bernstein operator, or its modified energy- and momentum-conserving counterpart. When approximating such operators numerically, it is important to respect their structure in order to satisfy the laws of thermodynamics. It is, however, challenging to discretise such operators in a structure-preserving way when considering particle methods. In this work, we present a macro-particle discretisation of the Lenard-Bernstein collision operator that is energy and momentum preserving.

physics.plasm-ph↗

arXiv2023

A metriplectic formulation of polarized radiative transfer

We present a metriplectic formulation of the radiative transfer equation with polarization and varying refractive index and show that this formulation automatically satisfies the first two laws of thermodynamics. In particular, the derived antisymmetric bracket enjoys the Jacobi identity. To obtain this formulation we suitably transform the equation and show that important physical quantities derived from the solution remain invariant under such a transformation.

math-ph↗

arXiv2022

Improved Perception of AEC Construction Details via Immersive Teaching in Virtual Reality

This work proposes, implements and tests an immersive framework upon Virtual Reality (VR) for comprehension, knowledge development and learning process assisting an improved perception of complex spatial arrangements in AEC in comparison to the traditional 2D projection drawing-based method. The research focuses on the prototypical example of construction details as a traditionally difficult teaching task for conveying geometric and semantic information to students. Our mixed-methods study analyses test results of two test panel groups upon different questions about geometric and functional aspects of the construction detail as well as surveys and interviews of participating lecturers, students and laypersons towards their experience using the VR tool. The quantitative analysis of the test results prove that for participants with little pre-existing knowledge (such as novice students), a significantly better learning score for the test group is detected. Moreover, both groups rated the VR experience as an enjoyable and engaging way of learning. Analysis of survey results towards the VR experience reveals, that students, lecturers and professionals alike enjoyed the VR experience more than traditional learning of the construction detail. During the post-experiment qualitative evaluation in the form of interviews, the panel expressed an improved understanding, increased enthusiasm for the topic, and greater desire for other topics to be presented using VR tools. The expressed better understanding of design concepts after the VR experience by the students is statistically significant on average in the exam results. The results support our core assumption, that the presentation of contextual 3D models is a promising teaching approach to illustrate content.

cs.HC↗

arXiv2022

Symplectic model reduction methods for the Vlasov equation

Particle-based simulations of the Vlasov equation typically require a large number of particles, which leads to a high-dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment.

physics.comp-ph↗

arXiv2022

Control of Parasitism in Variational Integrators for Degenerate Lagrangian Systems

This paper deals with the control of parasitism in variational integrators for degenerate Lagrangian systems by writing them as general linear methods. This enables us to calculate their parasitic growth parameters which are responsible for the loss of long-time energy conservation properties of these algorithms. As a remedy and to offset the effects of parasitism, the standard projection technique is then applied to the general linear methods to numerically preserve the invariants of the degenerate Lagrangian systems by projecting the solution onto the desired manifold.

math.NA↗

arXiv2021

Multi-Objective Loss Balancing for Physics-Informed Deep Learning

Physics-Informed Neural Networks (PINN) are algorithms from deep learning leveraging physical laws by including partial differential equations together with a respective set of boundary and initial conditions as penalty terms into their loss function. In this work, we observe the significant role of correctly weighting the combination of multiple competitive loss functions for training PINNs effectively. To this end, we implement and evaluate different methods aiming at balancing the contributions of multiple terms of the PINNs loss function and their gradients. After reviewing of three existing loss scaling approaches (Learning Rate Annealing, GradNorm and SoftAdapt), we propose a novel self-adaptive loss balancing scheme for PINNs named \emph{ReLoBRaLo} (Relative Loss Balancing with Random Lookback). We extensively evaluate the performance of the aforementioned balancing schemes by solving both forward as well as inverse problems on three benchmark PDEs for PINNs: Burgers' equation, Kirchhoff's plate bending equation and Helmholtz's equation. The results show that ReLoBRaLo is able to consistently outperform the baseline of existing scaling methods in terms of accuracy, while also inducing significantly less computational overhead.

cs.LG↗

arXiv2021

Struct-MRT: Immersive Learning and Teaching of Design and Verification in Structural Civil Engineering using Mixed Reality

Our goal is to transform traditional paper-based instruction into an immersive lesson. This paper presents the conception, workflow and deployment of two MR applications for verification of typical yet geometrically complex structural members: a reinforced concrete corbel and a steel frame. The aim of this research is threefold: (i) to develop and implement the technological feasibility of such applications, (ii) to demonstrate possible use cases in the context of structural engineering lectures and (iii) to evaluate the presented MR examples and the future potential of such MR applications in structural engineering lectures through a survey. The workflow and MR teaching applications were developed with Apple's ARKit. The verification process was reproduced in the MR applications based on conventional exercises taught on paper. Users can navigate independently through the applications and review every single step, including a true-to-scale, spatial representation of the specific component as well as associated verification formulas in the respective step. The applications were used to assess the demand and expectations for immersive teaching techniques among students and instructors through a survey. The participants were asked to test the MR applications on their devices or watch pre-recorded video demonstrations, afterwards perception was elicited through a questionnaire. The results of subsequent data analysis show generally positive judgement of the MR application over the six questioned categories (style, usefulness, ease of use, enjoyment, attitude as well as intention towards using). The statistical analysis revealed (positivity) biases for users with prior XR experience w.r.t. to usage and navigation, while inexperienced users underlined increased enjoyment or excitement with this learning format. The outlook covers identified shortcomings and future developments in this field.

cs.HC↗

arXiv2020

Variational integrators for stochastic dissipative Hamiltonian systems

Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange-d'Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether's theorem. Furthermore, mean-square and weak Lagrange-d'Alembert Runge-Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to non-geometric methods. The Vlasov-Fokker-Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

math.NA↗

arXiv2018

Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems

This paper explores energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator. We discuss two approaches, namely direct Galerkin formulations and discretizations of the underlying infinite-dimensional metriplectic structure of the collision integral. The spatial discretizations are chosen to reproduce the time-continuous conservation laws that correspond to Casimir invariants and to guarantee the positivity of the distribution function. Both the direct and the metriplectic discretization are demonstrated to have exact H-theorems and unique, physically exact equilibrium states. Most importantly, the two approaches are shown to coincide, given the chosen Galerkin method. A temporal discretization, preserving all of the mentioned properties, is achieved with so-called discrete gradients. Hence the proposed algorithm successfully translates all properties of the infinite-dimensional time-continuous Landau collision operator to time- and space-discrete sparse-matrix equations suitable for numerical simulation.

physics.plasm-ph↗

arXiv2018

Variational Integrators for Inertial Magnetohydrodynamics

Recently, an extended version of magnetohydrodynamics that incorporates electron inertia, dubbed inertial magnetohydrodynamics, has been proposed. This model features a noncanonical Hamiltonian formulation with a number of conserved quantities, including the total energy and modified versions of magnetic and cross helicity. In this work, a variational integrator is presented which preserves these conservation laws to machine accuracy. As long as effects due to finite electron mass are neglected, the scheme preserves the magnetic field line topology so that unphysical reconnection is absent. Only when effects of finite electron mass are added, magnetic reconnection takes place. The excellent conservation properties of the method are illustrated by numerical examples in 2D.

physics.comp-ph↗

arXiv2018

Metriplectic particle-in-cell integrators for the Landau collision operator

In this paper, we present a new framework for addressing the nonlinear Landau collision operator in terms of particle-in-cell methods. We employ the underlying metriplectic structure of the collision operator and, using a macro particle discretization for the distribution function, we transform the infinite-dimensional system into a finite-dimensional time-continuous metriplectic system for advancing the macro particle weights. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of density, momentum, and energy, as well as the positive semi-definite production of entropy in both the time-continuous and the fully discrete system is demonstrated algebraically. The new algorithm is fully compatible with the existing particle-in-cell Poisson integrators for the Vlasov-Maxwell system.

physics.comp-ph↗

arXiv2018

Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories

Symplectic integrators offer many advantages for the numerical solution of Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two of the central Hamiltonian systems encountered in plasma physics --- the flow of magnetic field lines and the guiding center motion of magnetized charged particles --- resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates, i.e., coordinates lacking the familiar $(q, p)$ partitioning. Recent efforts made progress toward non-canonical symplectic integration of these systems by appealing to the variational integration framework; however, those integrators were multistep methods and later found to be numerically unstable due to parasitic mode instabilities. This work eliminates the multistep character and, therefore, the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem ``degenerate variational integration''. Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that their resultant Euler-Lagrange equations are systems of first-order ODEs. We show that retaining the same degree of degeneracy when constructing a discrete Lagrangian yields one-step variational integrators preserving a non-canonical symplectic structure on the original Hamiltonian phase space. The advantages of the new algorithms are demonstrated via numerical examples, demonstrating superior stability compared to existing variational integrators for these systems and superior qualitative behavior compared to non-conservative algorithms.

physics.plasm-ph↗

arXiv2017

Projected Variational Integrators for Degenerate Lagrangian Systems

We propose and compare several projection methods applied to variational integrators for degenerate Lagrangian systems, whose Lagrangian is of the form $L = \vartheta(q) \cdot \dot{q} - H(q)$ and thus linear in velocities. While previous methods for such systems only work reliably in the case of $\vartheta$ being a linear function of $q$, our methods are long-time stable also for systems where $\vartheta$ is a nonlinear function of $q$. We analyse the properties of the resulting algorithms, in particular with respect to the conservation of energy, momentum maps and symplecticity. In numerical experiments, we verify the favourable properties of the projected integrators and demonstrate their excellent long-time fidelity. In particular, we consider a two-dimensional Lotka-Volterra system, planar point vortices with position-dependent circulation and guiding centre dynamics.

math.NA↗