Mini-data-driven Deep Arbitrary Polynomial Chaos Expansion for Uncertainty Quantification

The surrogate model-based uncertainty quantification method has drawn a lot of attention in recent years. Both the polynomial chaos expansion (PCE) and the deep learning (DL) are powerful methods for building a surrogate model. However, the 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 the DL also needs a lot of labeled data to train the neural network model. This paper proposes a deep arbitrary polynomial chaos expansion (Deep aPCE) method to improve the balance between surrogate model accuracy and training data cost. On the one hand, the multilayer perceptron (MLP) model is used to solve the adaptive expansion coefficients of arbitrary polynomial chaos expansion, which can improve the Deep aPCE model accuracy with lower expansion order. On the other hand, the adaptive arbitrary polynomial chaos expansion's properties are used to construct the MLP training cost function based on only a small amount of labeled data and a large scale of non-labeled data, which can significantly reduce the training data cost. Four numerical examples and an actual engineering problem are used to verify the effectiveness of the Deep aPCE method.