We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate polynomial function. The metamodel obtained enables a reliable estimation of the statistics of the output, provided that a suitable probabilistic model of the input is available. Machine learning (ML) regression is a research field that focuses on providing purely data-driven input-output maps, with the focus on pointwise prediction accuracy. We show that a PCE metamodel purely trained on data can yield pointwise predictions whose accuracy is comparable to that of other ML regression models, such as neural networks and support vector machines. The comparisons are performed on benchmark datasets available from the literature. The methodology also enables the quantification of the output uncertainties, and is robust to noise. Furthermore, it enjoys additional desirable properties, such as good performance for small training sets and simplicity of construction, with only little parameter tuning required.
National University of Defense Technology（国防科技大学）；Chinese Academy of Military Science（中国人民解放军军事科学院）
The surrogate model-based uncertainty quantification method has drawn much attention in many engineering fields. Polynomial chaos expansion (PCE) and deep learning (DL) are powerful methods for building a surrogate model. However, PCE needs to increase the expansion order to improve the accuracy of the surrogate model, which causes more labeled data to solve the expansion coefficients, and DL also requires a lot of labeled data to train the deep neural network (DNN). First of all, this paper proposes the adaptive arbitrary polynomial chaos (aPC) and proves two properties about the adaptive exp…
Uncertainty Quantification for Machine Learning-Based Prediction: A Polynomial Chaos Expansion Approach for Joint Model and Input Uncertainty Propagation
School of Mechanical Engineering（机械工程学院）；Purdue University（普渡大学）
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Technische Universität Munchen（慕尼黑工业大学）
The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings usin…
Model predictive control is an advanced control approach for multivariable systems with constraints, which is reliant on an accurate dynamic model. Most real dynamic models are however affected by uncertainties, which can lead to closed-loop performance deterioration and constraint violations. In this paper we introduce a new algorithm to explicitly consider time-invariant stochastic uncertainties in optimal control problems. The difficulty of propagating stochastic variables through nonlinear functions is dealt with by combining Gaussian processes with polynomial chaos expansions. The main no…
Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram-Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding co…
Parameter Estimation in Stochastic Differential Equations via Wiener Chaos Expansion and Stochastic Gradient Descent
Universitat Autónoma de Barcelona, Departament de Matemàtiques（巴塞罗那自治大学数学系）；Facultad de Ciencias - Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Secihti - Centro Interdisciplinario de Ciencias Marinas. IPN.（国家理工学院海洋科学跨学科中心）
Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDI…
Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors ("basis") for PCE adaptively. The goal of this paper is to help practitioners identify the most suitable methods for constructing a surrogate PCE for their model. We describe three state-of-the-art basis-adaptive a…
Gradient-enhanced global sensitivity analysis with Poincar{é} chaos expansions
UMR CNRS 5219, Institut de Mathématiques de Toulouse, INSA, Université de Toulouse（图卢兹数学研究所）；Chair of Risk, Safety and Uncertainty Quantification, ETH Zürich（苏黎世联邦理工学院风险、安全与不确定性量化讲席）
