A feature mapping strategy of metamodelling for nonlinear stochastic dynamical systems with low to high-dimensional input uncertainties

This paper deals with the issue on metamodelling (a.k.a. surrogate modelling) of nonlinear stochastic dynamical systems, which are often with multiple input uncertainties Θ∈Rⁿ, viz., the dimension n may range from low to high (e.g., n≥10). In this paper, to circumvent the problem of “curse of dimensionality” of high-dimensional input uncertainties, the feature spaces of outputs and inputs are firstly extracted from the original output and input spaces, and thus a feature mapping strategy is proposed. To form the feature output space, the nonlinear autoregressive with exogenous inputs (NARX) and the proper orthogonal decomposition (POD) are adopted, while the feature input space is detected by the active subspace method (ASM). It is found that the dimension of feature input (output) space may be much less than the one of original input (output) space, thus the applicability of many metamodelling methods can be naturally enhanced. On the constructed input–output feature space, the procedure of metamodelling is completed by the polynomial chaos expansion (PCE) combined with Kriging, which can capture global behaviours as well as local characteristics of the computational model. Two techniques are introduced to accelerate the proposed feature mapping strategy, consisting of the GF-discrepancy minimization algorithm for the design of experiments (DoEs), and the manifold optimization technique for the parameter identification of ASM. Four benchmarks, including a mathematical function (n=2), a dynamical quarter car model (n=10), a Bouc–Wen nonlinear oscillator subjected to earthquake ground motions (n=30), and the first sub-system (as a black box) of the NASA UQ Challenge 2019 (n=100), are studied to demonstrate the accuracy and efficiency of the proposed method. Some problems to be further studied are also outlined.