Multivariate sensitivity-adaptive polynomial chaos expansion for high-dimensional surrogate modeling and uncertainty quantification

TL;DR

The paper tackles the challenge of constructing accurate surrogates for vector-valued responses under high-dimensional inputs by introducing a multivariate sensitivity-adaptive polynomial chaos expansion (MVSA-PCE). It builds a common anisotropic polynomial basis via multivariate sensitivity indices and forward-neighbor enrichment within downward-closed index sets, enabling vectorized regression and efficient uncertainty quantification. Across engineering test cases (beam deflection, induction motor start-up, and power-grid load flow), MVSA-PCE delivers high accuracy with substantially lower data and computational demands than fixed-basis or element-wise sparse/adaptive methods, and remains applicable when outputs are thousands-strong. The approach supports online surrogate modeling and digital-twin applications, with potential extensions to multi-element, active learning, and physics-informed surrogates.

Abstract

This work develops a novel basis-adaptive method for constructing anisotropic polynomial chaos expansions of multidimensional (vector-valued, multi-output) model responses. The adaptive basis selection is based on multivariate sensitivity analysis metrics that can be estimated by post-processing the polynomial chaos expansion and results in a common anisotropic polynomial basis for the vector-valued response. This allows the application of the method to problems with up to moderately high-dimensional model inputs (in the order of tens) and up to very high-dimensional model responses (in the order of thousands). The method is applied to different engineering test cases for surrogate modeling and uncertainty quantification, including use cases related to electric machine and power grid modeling and simulation, and is found to produce highly accurate results with comparatively low data and computational demand.

Figures (12)

• Figure 1: (a) Sketch of the simply supported beam model. (b) Beam deflection for $10$ realizations of the simply supported beam model parameters.
• Figure 2: Simply supported beam: over model response of the different for response dimension $M \in \left\{10, 100, 1000\right\}$ and training data set size $Q \in \left\{50, 100, 150\right\}$. The solid lines show average error values over $10$ different training and test data sets. The shaded areas represent the difference between minimum and maximum errors over these $10$ data sets.
• Figure 3: Simply supported beam: Mean and standard deviation of the model's response with dimension $M=1000$, estimated with the different for training data set size $Q \in \left\{50, 100, 150\right\}$. The reference mean and standard deviation are computed via with $10^5$ random samples.
• Figure 4: Simply supported beam: Computation time of the different for response dimension $M \in \left\{10 ,100, 1000\right\}$ and training data set size $Q \in \left\{50 ,100, 150\right\}$. The colored bars show the average computation time over $10$ different training data sets. The black error bars show the difference between minimum and maximum computation time over these $10$ data sets.
• Figure 5: (a) Equivalent circuit model of an induction motor. (b) Electromagnetic torque generated during the start-up of the induction motor for $10$ realizations of the model parameters.