Zimmering, Bernd

Forecasting the behaviour of industrial robots, power grids or pandemics under changing external inputs requires accurate dynamical models that can adapt to varying signals and capture long-term effects such as delays or memory. While recent neural approaches address some of these challenges individually, their reliance on computationally intensive solvers and their black-box nature limit their practical utility. In this work, we propose Laplace-Net, a decoupled, solver-free neural framework for learning forced and delay-aware dynamical systems. It uses the Laplace transform to (i) bypass computationally intensive solvers, (ii) enable the learning of delays and memory effects and (iii) decompose each system into interpretable control-theoretic components. Laplace-Net also enhances transferability, as its modular structure allows for targeted re-training of individual components to new system setups or environments. Experimental results on eight benchmark datasets–including linear, nonlinear and delayed systems–demonstrate the method’s improved accuracy and robustness compared to state-of-the-art approaches, particularly in handling complex and previously unseen inputs.

This study presents an approach based on data-driven methods for determining the parameters needed to model time-dependent material behaviour. The time-dependent behaviour of the thermoplastic polymer polybutylene terephthalate is investigated. The material was examined under two conditions, one with and one without the inclusion of reinforcing short fibres. Two modelling approaches are proposed to represent the time-dependent response. The first approach is the generalised Maxwell model formulated through the classical exponential Prony series, and the second approach is a model based on fractional calculus. In order to quantify the comparative capabilities of both models, experimental data from tensile creep tests on fibre-reinforced polybutylene terephthalate and unreinforced polybutylene terephthalate specimens are analysed. A central contribution of this work is the implementation of a machine-learning-ready parameter identification framework that enables the automated extraction of model parameters directly from time-series data. This framework enables the robust fitting of the Prony-based model, which requires multiple characteristic times and stiffness parameters, as well as the fractional model, which achieves high accuracy with significantly fewer parameters. The fractional model benefits from a novel neural solver for fractional differential equations, which not only reduces computational complexity but also permits the interpretation of the fractional order and stiffness coefficient in terms of physical creep resistance. The methodological framework is validated through a comparative assessment of predictive performance, parameter cheapness, and interpretability of each model, thereby providing a comprehensive understanding of their applicability to long-term material behaviour modelling in polymer-based composite materials.

This work examines dynamic models for describing the viscoelastic behaviour of short-fibre reinforced plastics in tensile tests. The creep behaviour of reinforced PBT GF30 compared to unreinforced PBT GF0 is investigated on the basis of experimental data. Two different modelling approaches are compared: a generalised Maxwell model based on the prony series and a model with fractional derivatives. The experimental data show that glass fibres significantly reduce the deformation under constant load, as they stiffen the polymer matrix and inhibit creep deformation. Parameters can be determined for both models using machine learning methods. However, the Prony-Maxwell based model requires three parameters to accurately represent the data, whereas the fractional model only requires two parameters. The results clearly show the advantages of fractional model for the description of the long time series behaviour: on the one hand, fewer parameters are required and on the other hand, additional knowledge can be gained through the interpretation of the parameters obtained. The experimental data as well a the open-source software developed to learn the model is published alongside this work.

Neural Ordinary Differential Equations (NODEs) are well-established architectures that fit an ODE, modelled by a neural network (NN), to data, effectively modelling complex dynamical systems. Recently, Neural Fractional Differential Equations (NFDEs) were proposed, inspired by NODEs, to incorporate non-integer order differential equations, capturing memory effects and long-range dependencies. In this work, we present an optimised implementation of the NFDE solver, achieving up to 570 times faster computations and up to 79 times higher accuracy. Additionally, the solver supports efficient multidimensional computations and batch processing. Furthermore, we enhance the experimental design to ensure a fair comparison of NODEs and NFDEs by implementing rigorous hyperparameter tuning and using consistent numerical methods. Our results demonstrate that for systems exhibiting fractional dynamics, NFDEs significantly outperform NODEs, particularly in extrapolation tasks on unseen time horizons. Although NODEs can learn fractional dynamics when time is included as a feature to the NN, they encounter difficulties in extrapolation due to reliance on explicit time dependence. The code is available at https://github.com/zimmer-ing/Neural-FDE

This paper examines the integration of Continuous-Time Neural Networks (CTNNs), including Neural ODEs, CDEs, Neural Laplace, and Neural Flows, into engineering practices, particularly in dynamical system modeling. We provide a detailed introduction to CTNNs, highlighting their underutilization in engineering despite similarities with traditional Ordinary Differential Equation (ODE) models. Through a comparative analysis with conventional engineering approaches, using a spring-mass-damper system as an example, we demonstrate both theoretical and practical aspects of CTNNs in engineering contexts. Our work underscores the potential of CTNNs to harmonize with traditional engineering methods, exploring their applications in Cyber- Physical Systems (CPS). Additionally, we review key open-source software tools for implementing CTNNs, aiming to facilitate their broader integration into engineering practices.