# PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty

arXiv: https://arxiv.org/abs/2202.05063 · HTML: https://mlsanity.com/paper/2202.05063/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 83 references that are in the map.
- First posted on arXiv 10 Feb 2022.

## Builds on (most cited first, 22 of 22)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Towards Bayesian Deep Learning: A Framework and Some Existing Methods](https://mlsanity.com/paper/1608.06884/) (2016)
- [Penalty Method for Inversion-Free Deep Bilevel Optimization](https://mlsanity.com/paper/1911.03432/) (2019)
- [A Variational Auto-Encoder Model for Stochastic Point Processes](https://mlsanity.com/paper/1904.03273/) (2019)
- [Data-driven polynomial chaos expansion for machine learning regression](https://mlsanity.com/paper/1808.03216/) (2018)
- [Polynomial-Chaos-based Kriging](https://mlsanity.com/paper/1502.03939/) (2015)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Uncertainty Quantification with Generative Models](https://mlsanity.com/paper/1910.10046/) (2019)
- [Analysis of stochastic Lanczos quadrature for spectrum approximation](https://mlsanity.com/paper/2105.06595/) (2021)
- [Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approach](https://mlsanity.com/paper/1812.06309/) (2018)
- [Metrics for Benchmarking and Uncertainty Quantification: Quality, Applicability, and a Path to Best Practices for Machine Learning in Chemistry](https://mlsanity.com/paper/2010.00110/) (2020)

## Built on by (most cited first, 5 of 5)

- [Bi-fidelity Variational Auto-encoder for Uncertainty Quantification](https://mlsanity.com/paper/2305.16530/) (2023)
- [Deep Polynomial Chaos Expansion](https://mlsanity.com/paper/2507.21273/) (2025)
- [The Deep Arbitrary Polynomial Chaos Neural Network or how Deep Artificial Neural Networks could benefit from Data-Driven Homogeneous Chaos Theory](https://mlsanity.com/paper/2306.14753/) (2023)
- [All Emulators are Wrong, Many are Useful, and Some are More Useful Than Others: A Reproducible Comparison of Computer Model Surrogates](https://mlsanity.com/paper/2512.09060/) (2025)
- [Interpretable epistemic uncertainty decomposition in sequential generative models via polynomial chaos surrogates](https://mlsanity.com/paper/2510.21523/) (2025)

## Abstract (arXiv)

Learning data representations under uncertainty is an important task that emerges in numerous scientific computing and data analysis applications. However, uncertainty quantification techniques are computationally intensive and become prohibitively expensive for high-dimensional data. In this study, we introduce a dimensionality reduction surrogate modeling (DRSM) approach for representation learning and uncertainty quantification that aims to deal with data of moderate to high dimensions. The approach involves a two-stage learning process: 1) employing a variational autoencoder to learn a low-dimensional representation of the input data distribution; and 2) harnessing polynomial chaos expansion (PCE) formulation to map the low dimensional distribution to the output target. The model enables us to (a) capture the system dynamics efficiently in the low-dimensional latent space, (b) learn under uncertainty, a representation of the data and a mapping between input and output distributions, (c) estimate this uncertainty in the high-dimensional data system, and (d) match high-order moments of the output distribution; without any prior statistical assumptions on the data. Numerical results are presented to illustrate the performance of the proposed method.