Styfen Schär

Last Name

Schär

First Name

Styfen

ORCID

0000-0001-6715-220X

Organisational unit

03962 - Sudret, Bruno / Sudret, Bruno

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Publications 1 - 10 of 15

Autoregressive surrogate modeling for dynamical systems
Schär, Styfen (2025)
Numerical simulations are essential tools for predicting the behavior of real-world systems. They support the design of robust and cost-efficient civil structures and therefore contribute to public safety. In practical applications, one must often account for uncertainty in the simulator inputs (e.g. material properties, geometry, or loading conditions) since it translates into uncertainty in the model response (deflections or stresses). Understanding how the input uncertainty affects the system response is the objective of uncertainty quantification (UQ). A standard approach in UQ is Monte Carlo simulation, which estimates output variability by evaluating many input scenarios. While conceptually simple, this method can be impractical when the computational cost of repeated simulations is high. To address this, surrogate models are used as fast mathematical approximations of the full simulation, built from a limited number of simulator runs. However, constructing accurate surrogates from limited data is challenging, especially for dynamical systems, where the response evolves over time according to complex physical laws. For example, in a bridge exposed to variable traffic and wind loads, small uncertainties in material properties or loading history can lead to significantly different deflection patterns as time progresses. The behavior of such systems is therefore often difficult to emulate accurately using conventional surrogate modeling techniques. A promising class of surrogates to model dynamical systems is the family of nonlinear autoregressive with exogenous inputs (NARX) models, which use past system states to predict future responses. This approach enforces the causality inherent in any dynamical system, thus improving long-term accuracy. Although effective in many settings, classical NARX models show limitations when applied to highly complex systems or when data is limited. This thesis advances NARX-based surrogate modeling for complex dynamical systems through three main contributions. First, we develop the manifold-NARX (mNARX) model, which enhances the expressiveness of classical NARX by hierarchically incorporating prior knowledge and auxiliary system data. It constructs structured sequences of simpler intermediate models, improving forecast stability and accuracy in data-limited settings. This makes mNARX particularly useful in engineering applications where such additional information is available. Second, we introduce functional-NARX (F-NARX), a formulation tailored to systems with long memory. Instead of using raw time lags, F-NARX works with temporal features and leverages the smoothness and continuity of the underlying physical process. This mitigates the curse of dimensionality, enhances generalization, and improves robustness to the time discretization of the data. Third, we present an automated framework that builds on the structure of F-NARX to construct mNARX models directly from data. This replaces the previously manual and heuristic-driven model-building process with a systematic, data-driven approach. By recursively identifying causal patterns in the data and organizing the surrogate accordingly, the resulting workflow is less labor-intensive and more scalable, making mNARX modeling more accessible in practice. The proposed methodologies are validated on a range of case studies, including analytical examples and complex engineering systems such as wind turbines and multi-story buildings. In all cases, the surrogate models demonstrate high accuracy, strong long-term predictive capabilities, and computational efficiency. This work therefore provides a practical foundation for surrogate modeling of dynamical systems and expands the possibilities for uncertainty quantification in many engineering applications.
Comparison of probabilistic structural reliability methods for ultimate limit state assessment of wind turbines
Wang, Hong; Gramstad, Odin; Schär, Styfen; et al. (2024)
Structural Safety
The probabilistic design of offshore wind turbines aims to ensure structural safety in a cost-effective way. This involves conducting structural reliability assessments for different design options and considering different structural responses. There are several structural reliability methods, and this paper will apply and compare different approaches in some simplified case studies. In particular, the well known environmental contour method will be compared to a more novel approach based on sequential sampling and Gaussian processes regression for an ultimate limit state case study on the maximum flapwise blade root bending moment. For one of the case studies, results will also be compared to results from a brute force simulation approach. Interestingly, the comparison is very different from the two case studies. In one of the cases the environmental contours method agrees well with the sequential sampling method but in the other, results vary considerably. Probably, this can be explained by the violation of some of the assumptions associated with the environmental contour approach, i.e. that the short-term variability of the response is large compared to the long-term variability of the environmental conditions. Results from this simple comparison study suggests that the sequential sampling method can be a robust and computationally effective approach for structural reliability assessment.
Surrogate modeling with functional nonlinear autoregressive models (F-NARX)
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2025)
Reliability Engineering & System Safety
We propose a novel functional approach to surrogate modeling of dynamical systems with exogenous inputs. This approach, named Functional Nonlinear AutoRegressive with eXogenous inputs (F-NARX), approximates the system response based on temporal features of the exogenous inputs and the system response. This marks a major step away from the discrete-time-centric approach of classical NARX models, which determines the relationship between selected time steps of the input/output time series. By modeling the system in a time-feature space, F-NARX takes advantage of the temporal smoothness of the process being modeled, providing more stable predictions and reducing the dependence of model performance on the discretization of the time axis. In this work, we introduce an F-NARX implementation based on principal component analysis and polynomial regression. To further improve prediction accuracy, we also introduce a modified hybrid least angle regression approach to identify a sparse model structure and minimize the expected forecast error, rather than the one-step-ahead prediction error. We investigate the behavior and capabilities of our F-NARX implementation on two case studies: an eight-story building under wind loading and a three-story steel frame under seismic loading. Our results demonstrate that F-NARX has several favorable properties that make it well-suited to surrogate modeling applications.
Automatic manifold identification for mNARX models
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2023)
Seismic fragility analysis using mNARX modelling
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2024)
Assessing the seismic vulnerability of civil structures is crucial for safeguarding human lives and ensuring the long-term functionality of essential infrastructure. Nevertheless, quantifying the capability of a structure to withstand a potentially large spectrum of seismic events still poses a significant challenge, due to the high level of uncertainty involved. In the current state-of-the art, the uncertainty in the occurrence and magnitude of seismic events is modelled through a statistical ground-motion model (SGMM), which is then propagated through detailed computational models of the structure under investigation. Still, conducting Monte Carlo simulation using an SGMM model is often unfeasible on complex structures due to the high computational costs associated, e.g. due to high-resolution finite-element modelling (FEM). To tackle this problem, surrogate models have emerged as computationally efficient proxies for FEM simulations. These models are trained on a relatively small dataset, typically a few hundred to a thousand FEM simulations, and focus on mapping SGMM parameters directly to scalar building performance metrics, such as maximum interstory drift or other damage measures. Traditional surrogate models may however struggle to capture the stochastic nature of SGMMs, which exhibit significant latent variability. In other words, to each set of SGMM parameters corresponds an infinite number of ground motions. Additionally, these surrogates often provide only selected scalar properties of the time-dependent structural responses, rather than their complete time history. To address these limitations, we propose to take advantage of the recently developed mNARX surrogate modelling strategy [1] to approximate the full history of the system response. mNARX offers two key advantages over traditional surrogates. First, it acts as an emulator for the full FEM, providing full-time history predictions, hence offering a deeper insight into the structural behavior. Second, it allows for incorporating prior knowledge of the physical system, through the construction of an exogenous input manifold, which results in exceptional data efficiency, significantly decreasing the training data needed with respect to traditional surrogates. To illustrate the effectiveness of mNARX in seismic fragility analysis, we present a case study involving a three-story steel frame simulated using the open-source software OpenSees. The structure is exposed to real earthquake data from the PEER ground motion database. Our results show that the mNARX surrogate accurately emulates the quantities like the interstory drift, even when trained on very small datasets. [1] Schär, S. et al. “Emulating the dynamics of complex systems using autoregressive models on manifolds (mNARX)”, Mech. Syst. Signal Proces., 2023
A data-driven surrogate model for uncertainty quantification of dynamical systems
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2023)
Surrogate models have become a standard tool in uncertainty quantification to emulate the response of real-world systems at a low computational cost. Within the HIPERWIND EU project, we recently proposed a class of surrogate models that can accurately model the time-dependent response of dynamical state- and control system-dependent structures excited by high-dimensional exogenous inputs, such as wind turbines (Dimitrov et al. (2022)). We achieved this by combining spectral compression strategies to handle the high spatial dimensionality of the inputs, with non-linear auto-regressive with exogenous input (NARX) models to capitalize on the temporal coherence of both inputs and outputs. In this novel approach called manifold NARX (mNARX) we build an input manifold consisting of the compressed exogenous input and several intermediate NARX models of problem-specific and modelling-relevant quantities. This manifold is then used as an exogenous input to a final NARX surrogate that is able to predict the target quantity of interest. This approach can emulate highly non-linear systems with simple, fast-to-train and fast-to-evaluate polynomial NARX base models. As a drawback of using multiple NARX models and dimensionality reduction steps, the space of tunable parameters in this surrogate modelling chain grows large. Consequently, despite the fast polynomial models, the process of model structure selection and parameter estimation can become laborious. In this contribution, we extend and partially automate the mNARX emulator design and training by first assembling a large set of potentially important features. We then capitalize on techniques from the machine learning literature to perform an automated feature selection and dimensionality reduction step. These techniques are applied with the goal of finding a suitable manifold that minimizes the error of the final NARX surrogate. Algorithmically, we adopt the DRSM (dimensionality reduction for surrogate modelling) framework originally developed in Lataniotis et al. (2020). DRSM is a state-of-the-art supervised algorithm that can jointly identify and train both non-linear dimensionality reduction techniques and surrogate models, with the goal to maximize surrogate modelling accuracy. Because DRSM is a data-driven framework that can be easily automated, it is ideal to circumvent the disadvantages of mNARX. We demonstrate that this extended mNARX can yield comparably or more accurate surrogates on a set of complex time-dependent benchmark systems, than the originally hand-tuned mNARX. Exemplary results of a wind turbine case study are shown in the figure. Further, we show that the extended mNARX remains fast despite the computationally intensive DRSM algorithm. By providing a more data-driven, less elaborate surrogate modelling procedure, we also push the application of mNARX toward other areas of research and industry. [1] Dimitrov, N., S. Marelli, and S. Schär (2022). Novel surrogate modelling approaches for wind turbine reliability assessment. H2020 Project HIPERWIND. Deliverable D4.1. [2] Lataniotis, C., S. Marelli, and B. Sudret (2020). Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approach. International Journal for Uncertainty Quantification 10 (1), 55–82.
Feature-centric NARX model with automated feature selection for the uncertainty quantification of dynamical systems
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2025)
Methods for efficient ULS reliability calculations and their impact on probabilistic design
Cousin, Alexis; Munoz-Zuniga, Miguel; Franceschini, Luca; et al. (2024)
Emulating the dynamics of complex systems using autoregressive models on manifolds (mNARX)
Schär, Styfen; Marelli, Stefano; Sudret, Bruno (2024)
Mechanical Systems and Signal Processing
We propose a novel surrogate modelling approach to efficiently and accurately approximate the response of complex dynamical systems driven by time-varying exogenous excitations over extended time periods. Our approach, namely manifold nonlinear autoregressive modelling with exogenous input (mNARX), involves constructing a problem-specific exogenous input manifold that is optimal for constructing autoregressive surrogates. The manifold, which forms the core of mNARX, is constructed incrementally by incorporating the physics of the system, as well as prior expert- and domain-knowledge. Because mNARX decomposes the full problem into a series of smaller sub-problems, each with a lower complexity than the original, it scales well with the complexity of the problem, both in terms of training and evaluation costs of the final surrogate. Furthermore, mNARX synergizes well with traditional dimensionality reduction techniques, making it highly suitable for modelling dynamical systems with high-dimensional exogenous inputs, a class of problems that is typically challenging to solve. Since domain knowledge is particularly abundant in physical systems, such as those found in civil and mechanical engineering, mNARX is well suited for these applications. We demonstrate that mNARX outperforms traditional autoregressive surrogates in predicting the response of a classical coupled spring-mass system excited by a one-dimensional random excitation. Additionally, we show that mNARX is well suited for emulating very high-dimensional time- and state-dependent systems, even when affected by active controllers, by surrogating the dynamics of a realistic aero-servo-elastic onshore wind turbine simulator. In general, our results demonstrate that mNARX offers promising prospects for modelling complex dynamical systems, in terms of accuracy and efficiency.
End-to-end wind turbine design under uncertainties: a practical example
Dimitrov, Nikolay K.; Kelly, Mark; McWilliam, Michael; et al. (2024)
Journal of Physics: Conference Series
This paper illustrates the process of design under uncertainty on a practical case study of an offshore wind farm. We document the entire process through selection and quantification of relevant uncertainties, definition of probabilistic limit states, reliability computation algorithms, as well as illustrating the impacts of the analysis through a design utilization study. The brief introduction in this study draws information and summarizes outcomes from the extensive works that took part within the EU H2020 HIPERWIND project. The results from the study show that significant material savings can be achieved by introducing probabilistic design methodologies, and particularly with the help of an integrated modelling approach where the entire structure (turbine, tower & foundation) is considered as a whole.

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Styfen Schär

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