Deep Adaptive Arbitrary Polynomial Chaos Expansion: A Mini-data-driven Semi-supervised Method for Uncertainty Quantification

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 expansion coefficients. Based on the adaptive aPC, a semi-supervised deep adaptive arbitrary polynomial chaos expansion (Deep aPCE) method is proposed to reduce the training data cost and improve the surrogate model accuracy. For one hand, the Deep aPCE method uses two properties of the adaptive aPC to assist in training the DNN based on only a small amount of labeled data and many 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 Deep aPCE model accuracy with lower expansion order. Besides, the Deep aPCE method can directly construct accurate surrogate models of the high dimensional stochastic systems without complex dimension-reduction and model decomposition operations. Five numerical examples and an actual engineering problem are used to verify the effectiveness of the Deep aPCE method.