Deep adaptive arbitrary polynomial chaos expansion: A mini-data-driven semi-supervised method for uncertainty quantification

All kinds of uncertainties influence the reliability of the engineering system. Thus, uncertainty quantification is significant to the system reliability analysis. Polynomial chaos expansion (PCE) is an effective method for uncertainty quantification while it requires sufficient labeled data to quantify uncertainty accurately. To overcome this problem, this paper proposes the adaptive arbitrary polynomial chaos (aPC) and proves two properties of the adaptive expansion coefficients. Sequentially, a semi-supervised deep adaptive arbitrary polynomial chaos expansion (Deep aPCE) method is proposed based on the adaptive aPC and the deep neural network (DNN). The Deep aPCE method uses two properties of the adaptive aPC to assist in training the DNN by a small amount of labeled data and abundant unlabeled data, significantly reducing the training data cost. On the other hand, the Deep aPCE method adopts the DNN to fine-tune the adaptive expansion coefficients dynamically, improving the accuracy of uncertainty quantification. Besides, the Deep aPCE method can directly construct accurate surrogate models of the high dimensional stochastic systems. Five numerical examples are used to verify the effectiveness of the Deep aPCE method. Finally, the Deep aPCE method is applied to the reliability analysis of an axisymmetric conical aircraft.