1 Introduction
In structural dynamics, experiments are necessary in order to determine the parameters of nonlinear systems, such as for the optimal design of dampers [
1] or within the scope of vibration-based identification [
2‐
4]. A significant effort has been spent on the development of experimental setups in civil engineering, such as using large-scale actuators and shaking tables. With real-time hybrid simulation, these experiments can also be combined with numerical models, such as for the investigation of soil–structure interaction [
5]. However, the existing physical testing procedures are limited to pure performance assessment and do not result in obtaining a model of the tested specimen.
Machine learning methods are getting increasingly popular for modeling of nonlinear dynamical systems and are implemented in various applications, such as anti-seismic materials and devices, cf. [
6] and the references therein. These models are suitable for fast and accurate modeling. However, a common bottleneck is the required amount and quality of data, which are necessary for the training of the machine learner. Specifically in structural dynamics applications, generation of real response data is hard and expensive to obtain which motivates developments of efficient data generation methods.
The need for more efficient data collection has been addressed by
design of experiments (DoE) where
unsupervised methods, such as random and Latin hypercube sampling (LHS) and sequential space-filling methods (e.g., Sobol sequence), are used [
7].
Active learning or adaptive sampling aims to further improve the data collection by considering the current performance of the model, based around some measure of informativeness, in order to gather data [
8]. There are numerous active learning strategies which can be employed based on the application [
9]. These methods are being researched for modern machine learning methods, such as the physics-informed neural networks (PINNs) for forward and inverse modeling. For instance, Nabian et al. proposed importance sampling to improve the training performance of PINNs [
10]. Further methods, including non-adaptive and residual-based adaptive sampling, have been studied by Wu et al. [
11]. While successful in their intended application, these sampling methods are developed for static mappings where the desired states can be directly collected, and the function evaluations are inexpensive. Hence, they are not for sampling of dynamic data and for applications where the cost of data acquisition is a limiting factor, as it is the case in this paper.
Another approach in active learning strategies dealing with identification and modeling of nonlinear dynamic systems is using uncertainty, which is a convenient byproduct of probabilistic machine learning methods, such as Gaussian Process (GP). For instance, Zhao et al. used active learning approach based on GP regression to reduce the number of dissipative particle dynamics simulations for multiscale modeling of non-Newtonian fluids [
12]. Despite its successful applications in active learning, GP is known for not scaling well with the amount of provided data. Similar to GP uncertainty, Belz et al. exploit special properties of local model networks (LMNs), namely local model errors, on which they base the active learning strategy and sample only the points which lay in the areas with high local errors [
13]. However, in our proposed framework, we assume no special properties coming from the model, such as local errors or uncertainty, are available.
In dynamic sampling problems, the system needs to be guided through unknown dynamics in order to explore the desired states. This is attempted by designing excitation signals, which drive the system to cover most of its operating range such as modulated chirp signal [
14] and binary signals, such as pseudorandom binary sequences (PRBS) for linear systems and amplitude modulated pseudorandom binary sequences (APRBS) for nonlinear systems [
15]. Extensions of these approaches belong to sequential signal design [
16] where the previously measured data are taken into account in order to boost diversity of the new data to be collected. However, these approaches belong to the batch methods and do not consider the current performance of the model. Hametner et al. addressed the identification of both static and dynamic nonlinear systems using the Fisher information matrix as an informativeness criteria [
17]. However, the Fisher information matrix requires an
a priori chosen model structure and its linearization, which are both problematic for nonlinear systems as pointed out by Nelles [
7]. Furthermore, data generation in dynamical systems has also been addressed in control applications by Buisson-Fenet et al. who proposed several algorithms for active learning using GPs [
18]. While the authors work provides a significant improvement over batch data generation methods, as mentioned above, our aim is to develop a framework which assumes that uncertainty information coming from the model is not available.
In some civil engineering applications, such as
structural health monitoring (SHM), the stream of data is available and needs to be selected efficiently for labeling and interpretation. Active learning strategies have been applied in SHM applications [
19‐
21] and for digital twins [
22]. However, the stream of data is not available for the considered problem setup of the present paper and our task is purposeful generation of data.
Despite the aforementioned research efforts and to the best of authors’ knowledge, a systematic data generation approach, such as testing-integrated modeling, has not been attempted in the domain of structural dynamics. Aiming to fill this gap, we introduce a theoretical framework for the adaptive data generation to allow automatic modeling of nonlinear dynamical systems, such as structural vibration control devices. The focus of the study is on systems where the specimen mass cannot be decoupled from the restoring force, such as tuned mass dampers (TMDs), hence requiring sampling of dynamic data. Optimized data sampling is coupled with modeling of the nonlinear restoring force using feedforward neural networks (FNNs) and carried out by the pattern search (PS) method. Numerical simulations are conducted to validate the proposed framework.
The paper is structured as follows. The modeling setup and the proposed data generation framework are introduced in Sect.
2. The numerical simulations and the comparison to other sampling methods are presented in Sect.
3 on a Duffing oscillator. An engineering application example is presented on a two-story shear frame with a nonlinear TMD. Finally, the contributions of this work are summarized in Sect.
4.
4 Conclusion
In this paper, we proposed an adaptive data generation framework for the testing-integrated modeling of nonlinear dynamic systems using machine learning. The focus of the study was on systems where the mass cannot be decoupled from the restoring force, hence requiring sampling of the dynamic data. The proposed framework yields an FNN model of nonlinear restoring force by sequentially evaluating the model performance on a given test data and providing a new set of excitation signal parameters in order to get more informative data in the next iteration. Accordingly, the collection of the dynamic data is converted to an optimization problem in a static sampling space defined by the excitation signal parameters. Hence, the main advantages of the proposed framework are the following:
-
The proposed framework converts the dynamic data sampling into a static optimization problem which enables the use of powerful conventional optimization algorithms.
-
The proposed framework is not tied to a specific machine learning model choice, allowing the user to choose the most suitable model class.
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By using a test set which covers the operating range of the modeled system, the final modeling error also provides the model confidence level. This assumption comes from the focus on structural dynamics applications, where such experiments are already being conducted for performance assessment, such as with vibration control devices.
The proposed framework was validated numerically on an example of Duffing oscillator and compared to three commonly used unsupervised sampling methods. It outperformed the unsupervised methods, having a success rate of finding the optimum training set twice as high. Furthermore, an engineering application was illustrated where the response of a two-story shear frame with a nonlinear tuned mass damper was simulated using the FNN model of the damper’s restoring force obtained both by the proposed framework and by the unsupervised methods. This example highlighted the capability of the proposed framework to extract the full information about the nonlinear response of the system.
The application of the proposed data generation framework to other testing configurations, such as damping testing systems, with direct restoring force identification, and the adoption of corresponding machine learning methods will be the focus of future research.
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