A data-driven surrogate model for uncertainty quantification of dynamical systems
Creators: name: Schär, Styfen
name: Marelli, Stefano
name: Sudret, Bruno
Publisher: name: ETH Zurich
Year: 2023

description: 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.