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

arXiv: https://arxiv.org/abs/2107.10428 · HTML: https://mlsanity.com/paper/2107.10428/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 90 references that are in the map.
- First posted on arXiv 22 Jul 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Deeply learned face representations are sparse, selective, and robust](https://mlsanity.com/paper/1412.1265/) (2014)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design](https://mlsanity.com/paper/1712.10131/) (2017)
- [Global sensitivity analysis in the context of imprecise probabilities (p-boxes) using sparse polynomial chaos expansions](https://mlsanity.com/paper/1705.10061/) (2017)
- [Infrastructure Recovery Curve Estimation Using Gaussian Process Regression on Expert Elicited Data](https://mlsanity.com/paper/2008.10211/) (2020)

## Built on by (most cited first, 6 of 6)

- [Multi-fidelity surrogate modeling for temperature field prediction using deep convolution neural network](https://mlsanity.com/paper/2301.06674/) (2023)
- [Deep Polynomial Chaos Expansion](https://mlsanity.com/paper/2507.21273/) (2025)
- [Consistency regularization-based Deep Polynomial Chaos Neural Network Method for Reliability Analysis](https://mlsanity.com/paper/2203.15655/) (2022)
- [Uncertainty Guided Ensemble Self-Training for Semi-Supervised Global Field Reconstruction](https://mlsanity.com/paper/2302.11940/) (2023)
- [Reliability Analysis of Complex Multi-State System Based on Universal Generating Function and Bayesian Network](https://mlsanity.com/paper/2208.04130/) (2022)
- [GenFormer: A Deep-Learning-Based Approach for Generating Multivariate Stochastic Processes](https://mlsanity.com/paper/2402.02010/) (2024)

## Abstract (arXiv)

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.