arXiv subjects

Jörg Fehr

Publications and source records attributed to Jörg Fehr.

16 recordsLinked to original sources

arXiv2025

VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification

The simulation of many complex phenomena in engineering and science requires solving expensive, high-dimensional systems of partial differential equations (PDEs). To circumvent this, reduced-order models (ROMs) have been developed to speed up computations. However, when governing equations are unknown or partially known, typically ROMs lack interpretability and reliability of the predicted solutions. In this work we present a data-driven, non-intrusive framework for building ROMs where the latent variables and dynamics are identified in an interpretable manner and uncertainty is quantified. Starting from a limited amount of high-dimensional, noisy data the proposed framework constructs an efficient ROM by leveraging variational autoencoders for dimensionality reduction along with a newly introduced, variational version of sparse identification of nonlinear dynamics (SINDy), which we refer to as Variational Identification of Nonlinear Dynamics (VINDy). In detail, the method consists of Variational Encoding of Noisy Inputs (VENI) to identify the distribution of reduced coordinates. Simultaneously, we learn the distribution of the coefficients of a pre-determined set of candidate functions by VINDy. Once trained offline, the identified model can be queried for new parameter instances and new initial conditions to compute the corresponding full-time solutions. The probabilistic setup enables uncertainty quantification as the online testing consists of Variational Inference naturally providing Certainty Intervals (VICI). In this work we showcase the effectiveness of the newly proposed VINDy method in identifying interpretable and accurate dynamical system for the Roessler system with different noise intensities and sources. Then the performance of the overall method - named VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics and fluid dynamics.

cs.LG↗

arXiv2024

Towards Using Active Learning Methods for Human-Seat Interactions To Generate Realistic Occupant Motion

In the context of developing new vehicle concepts, especially autonomous vehicles with novel seating arrangements and occupant activities, predicting occupant motion can be a tool for ensuring safety and comfort. In this study, a data-driven surrogate contact model integrated into an optimal control framework to predict human occupant behavior during driving maneuvers is presented. High-fidelity finite element simulations are utilized to generate a dataset of interaction forces and moments for various human body configurations and velocities. To automate the generation of training data, an active learning approach is introduced, which iteratively queries the high-fidelity finite element simulation for an additional dataset. The feasibility and effectiveness of the proposed method are demonstrated through a case study of a head interaction with an automotive headrest, showing promising results in accurately replicating contact forces and moments while reducing manual effort.

cs.RO↗

arXiv2024

Data-driven identification of latent port-Hamiltonian systems

Conventional physics-based modeling techniques involve high effort, e.g., time and expert knowledge, while data-driven methods often lack interpretability, structure, and sometimes reliability. To mitigate this, we present a data-driven system identification framework that derives models in the port-Hamiltonian (pH) formulation. This formulation is suitable for multi-physical systems while guaranteeing the useful system theoretical properties of passivity and stability. Our framework combines linear and nonlinear reduction with structured, physics-motivated system identification. In this process, high-dimensional state data obtained from possibly nonlinear systems serves as input for an autoencoder, which then performs two tasks: (i) nonlinearly transforming and (ii) reducing this data onto a low-dimensional latent space. In this space, a linear pH system, that satisfies the pH properties per construction, is parameterized by the weights of a neural network. The mathematical requirements are met by defining the pH matrices through Cholesky factorizations. The neural networks that define the coordinate transformation and the pH system are identified in a joint optimization process to match the dynamics observed in the data while defining a linear pH system in the latent space. The learned, low-dimensional pH system can describe even nonlinear systems and is rapidly computable due to its small size. The method is exemplified by a parametric mass-spring-damper and a nonlinear pendulum example, as well as the high-dimensional model of a disc brake with linear thermoelastic behavior.

math.DS↗

arXiv2024

Error Analysis of Randomized Symplectic Model Order Reduction for Hamiltonian systems

Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters and show that with a proper choice of hyperparameters, the projection error of rcSVD is at most a constant factor worse than the projection error of cSVD. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.

math.NA↗

arXiv2024

On using Machine Learning Algorithms for Motorcycle Collision Detection

Globally, motorcycles attract vast and varied users. However, since the rate of severe injury and fatality in motorcycle accidents far exceeds passenger car accidents, efforts have been directed toward increasing passive safety systems. Impact simulations show that the risk of severe injury or death in the event of a motorcycle-to-car impact can be greatly reduced if the motorcycle is equipped with passive safety measures such as airbags and seat belts. For the passive safety systems to be activated, a collision must be detected within milliseconds for a wide variety of impact configurations, but under no circumstances may it be falsely triggered. For the challenge of reliably detecting impending collisions, this paper presents an investigation towards the applicability of machine learning algorithms. First, a series of simulations of accidents and driving operation is introduced to collect data to train machine learning classification models. Their performance is henceforth assessed and compared via multiple representative and application-oriented criteria.

cs.LG↗

arXiv2024

Multi-Hierarchical Surrogate Learning for Structural Dynamical Crash Simulations Using Graph Convolutional Neural Networks

Crash simulations play an essential role in improving vehicle safety, design optimization, and injury risk estimation. Unfortunately, numerical solutions of such problems using state-of-the-art high-fidelity models require significant computational effort. Conventional data-driven surrogate modeling approaches create low-dimensional embeddings for evolving the dynamics in order to circumvent this computational effort. Most approaches directly operate on high-resolution data obtained from numerical discretization, which is both costly and complicated for mapping the flow of information over large spatial distances. Furthermore, working with a fixed resolution prevents the adaptation of surrogate models to environments with variable computing capacities, different visualization resolutions, and different accuracy requirements. We thus propose a multi-hierarchical framework for structurally creating a series of surrogate models for a kart frame, which is a good proxy for industrial-relevant crash simulations, at different levels of resolution. For multiscale phenomena, macroscale features are captured on a coarse surrogate, whereas microscale effects are resolved by finer ones. The learned behavior of the individual surrogates is passed from coarse to finer levels through transfer learning. In detail, we perform a mesh simplification on the kart model to obtain multi-resolution representations of it. We then train a graph-convolutional neural network-based surrogate that learns parameter-dependent low-dimensional latent dynamics on the coarsest representation. Subsequently, another, similarly structured surrogate is trained on the residual of the first surrogate using a finer resolution. This step can be repeated multiple times. By doing so, we construct multiple surrogates for the same system with varying hardware requirements and increasing accuracy.

cs.LG↗

arXiv2023

Dynamic Human Body Models in Vehicle Safety: An Overview

Significant trends in the vehicle industry are autonomous driving, micromobility, electrification and the increased use of shared mobility solutions. These new vehicle automation and mobility classes lead to a larger number of occupant positions, interiors and load directions. As safety systems interact with and protect occupants, it is essential to place the human, with its variability and vulnerability, at the center of the design and operation of these systems. Digital human body models (HBMs) can help meet these requirements and are therefore increasingly being integrated into the development of new vehicle models. This contribution provides an overview of current HBMs and their applications in vehicle safety in different driving modes. The authors briefly introduce the underlying mathematical methods and present a selection of HBMs to the reader. An overview table with guideline values for simulation times, common applications and available variants of the models is provided. To provide insight into the broad application of HBMs, the authors present three case studies in the field of vehicle safety: (i) in-crash finite element simulations and injuries of riders on a motorcycle; (ii) scenario-based assessment of the active pre-crash behavior of occupants with the Madymo multibody HBM; (iii) prediction of human behavior in a take-over scenario using the EMMA model.

math.DS↗

arXiv2023

Improved a posteriori Error Bounds for Reduced port-Hamiltonian Systems

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a second approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are for the first time adapted to port-Hamiltonian systems by exploiting their structure. The mathematical relationship between the two methods is discussed both, theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure interaction.

math.NA↗

arXiv2023

Randomized Symplectic Model Order Reduction for Hamiltonian Systems

Simulations of large scale dynamical systems in multi-query or real-time contexts require efficient surrogate modelling techniques, as e.g. achieved via Model Order Reduction (MOR). Recently, symplectic methods like the complex singular value decomposition (cSVD) or the SVD-like decomposition have been developed for preserving Hamiltonian structure during MOR. In the current contribution, we show how symplectic structure preserving basis generation can be made more efficient with randomized matrix factorizations. We present a randomized complex SVD (rcSVD) algorithm and a randomized SVD-like (rSVD-like) decomposition. We demonstrate the efficiency of the approaches with numerical experiments on high dimensional systems.

math.NA↗

arXiv2023

Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reduction

In recent decades, the main focus of computer modeling has been on supporting the design and development of engineering prototyes, but it is now ubiquitous in non-traditional areas such as medical rehabilitation. Conventional modeling approaches like the finite element~(FE) method are computationally costly when dealing with complex models, making them of limited use for purposes like real-time simulation or deployment on low-end hardware, if the model at hand cannot be simplified in a useful manner. Consequently, non-traditional approaches such as surrogate modeling using data-driven model order reduction are used to make complex high-fidelity models more widely available anyway. They often involve a dimensionality reduction step, in which the high-dimensional system state is transformed onto a low-dimensional subspace or manifold, and a regression approach to capture the reduced system behavior. While most publications focus on one dimensionality reduction, such as principal component analysis~(PCA) (linear) or autoencoder (nonlinear), we consider and compare PCA, kernel PCA, autoencoders, as well as variational autoencoders for the approximation of a structural dynamical system. In detail, we demonstrate the benefits of the surrogate modeling approach on a complex FE model of a human upper-arm. We consider both the models deformation and the internal stress as the two main quantities of interest in a FE context. By doing so we are able to create a computationally low cost surrogate model which captures the system behavior with high approximation quality and fast evaluations.

cs.LG↗

arXiv2022

Identification of Friction Models for MPC-based Control of a PowerCube Serial Robot

For model-based control, an accurate and in its complexity suitable representation of the real system is a decisive prerequisite for high and robust control quality. In a structured step-by-step procedure, a model predictive control (MPC) scheme for a Schunk PowerCube robot is derived. Neweul-M$^2$ provides the necessary nonlinear model in symbolical and numerical form. To handle the heavy online computational burden involved with the derived nonlinear model, a linear time-varying MPC scheme is developed based on linearizing the nonlinear system concerning the desired trajectory and the a priori known corresponding feed-forward controller. To improve the identification of the nonlinear friction models of the joints, a nonlinear regression method and the Sparse Identification of Nonlinear Dynamics (SINDy) are compared with each other concerning robustness, online adaptivity, and necessary preprocessing of the input data. Everything is implemented on a slim, low-cost control system with a standard laptop PC.

eess.SY↗

arXiv2022

Port-Hamiltonian Fluid-Structure Interaction Modeling and Structure-Preserving Model Order Reduction of a Classical Guitar

A fluid-structure interaction model in a port-Hamiltonian representation is derived for a classical guitar. We combine the laws of continuum mechanics for solids and fluids within a unified port-Hamiltonian (pH) modeling approach by adapting the discretized equations on second-order level in order to obtain a damped multi-physics model. The high-dimensionality of the resulting system is reduced by model order reduction. The article focuses on pH-systems in different state transformations, a variety of basis generation techniques as well as structure-preserving model order reduction approaches that are independent from the projection basis. As main contribution a thorough comparison of these method combinations is conducted. In contrast to typical frequency-based simulations in acoustics, transient time simulations of the system are presented. The approach is embedded into a straightforward workflow of sophisticated commercial software modeling and flexible in-house software for multi-physics coupling and model order reduction.

math.DS↗

arXiv2021

Real-time Human Response Prediction Using a Non-intrusive Data-driven Model Reduction Scheme

Recent research in non-intrusive data-driven model order reduction (MOR) enabled accurate and efficient approximation of parameterized ordinary differential equations (ODEs). However, previous studies have focused on constant parameters, whereas time-dependent parameters have been neglected. The purpose of this paper is to introduce a novel two-step MOR scheme to tackle this issue. In a first step, classic MOR approaches are applied to calculate a low-dimensional representation of high-dimensional ODE solutions, i.e. to extract the most important features of simulation data. Based on this representation, a long short-term memory (LSTM) is trained to predict the reduced dynamics iteratively in a second step. This enables the parameters to be taken into account during the respective time step. The potential of this approach is demonstrated on an occupant model within a car driving scenario. The reduced model's response to time-varying accelerations matches the reference data with high accuracy for a limited amount of time. Furthermore, real-time capability is achieved. Accordingly, it is concluded that the presented method is well suited to approximate parameterized ODEs and can handle time-dependent parameters in contrast to common methods.

math.DS↗

arXiv2021

Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators

We discuss nonlinear model predictive control (NMPC) for multi-body dynamics via physics-informed machine learning methods. Physics-informed neural networks (PINNs) are a promising tool to approximate (partial) differential equations. PINNs are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus present the idea of enhancing PINNs by adding control actions and initial conditions as additional network inputs. The high-dimensional input space is subsequently reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms. Finally, we present our results using our PINN-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator.

math.OC↗

arXiv2020

Sustainable Research Software Hand-Over

Scientific software projects evolve rapidly in their initial development phase, yet at the end of a funding period, the completion of a research project, thesis, or publication, further engagement in the project may slow down or cease completely. To retain the invested effort for the sciences, this software needs to be preserved or handed over to a succeeding developer or team, such as the next generation of (PhD) students. Comparable guides provide top-down recommendations for project leads. This paper intends to be a bottom-up approach for sustainable hand-over processes from a developer's perspective. An important characteristic in this regard is the project's size, by which this guideline is structured. Furthermore, checklists are provided, which can serve as a practical guide for implementing the proposed measures.

cs.GL↗

arXiv2016

Best Practices for Replicability, Reproducibility and Reusability of Computer-Based Experiments Exemplified by Model Reduction Software

Over the recent years the importance of numerical experiments has gradually been more recognized. Nonetheless, sufficient documentation of how computational results have been obtained is often not available. Especially in the scientific computing and applied mathematics domain this is crucial, since numerical experiments are usually employed to verify the proposed hypothesis in a publication. This work aims to propose standards and best practices for the setup and publication of numerical experiments. Naturally, this amounts to a guideline for development, maintenance, and publication of numerical research software. Such a primer will enable the replicability and reproducibility of computer-based experiments and published results and also promote the reusability of the associated software.

cs.MS↗