# Deep Polynomial Chaos Expansion

arXiv: https://arxiv.org/abs/2507.21273 · HTML: https://mlsanity.com/paper/2507.21273/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 97 references that are in the map.
- First posted on arXiv 28 Jul 2025.

## Builds on (most cited first, 28 of 28)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Scaling Up Probabilistic Circuits by Latent Variable Distillation](https://mlsanity.com/paper/2210.04398/) (2022)
- [Adaptive wavelet distillation from neural networks through interpretations](https://mlsanity.com/paper/2107.09145/) (2021)
- [Data-driven polynomial chaos expansion for machine learning regression](https://mlsanity.com/paper/1808.03216/) (2018)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Polynomial-Chaos-based Kriging](https://mlsanity.com/paper/1502.03939/) (2015)
- [Visualizing and Understanding Sum-Product Networks](https://mlsanity.com/paper/1608.08266/) (2016)
- [Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models](https://mlsanity.com/paper/2107.09814/) (2021)
- [Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/2108.05623/) (2021)
- [Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications](https://mlsanity.com/paper/2009.04800/) (2020)
- [A Guide to Feature Importance Methods for Scientific Inference](https://mlsanity.com/paper/2404.12862/) (2024)
- [Bayesian Surrogate Analysis and Uncertainty Propagation](https://mlsanity.com/paper/2101.04038/) (2021)
- [PCE-PINNs: Physics-Informed Neural Networks for Uncertainty Propagation in Ocean Modeling](https://mlsanity.com/paper/2105.02939/) (2021)
- [Bayesian Structure Scores for Probabilistic Circuits](https://mlsanity.com/paper/2302.12130/) (2023)
- [Deep Adaptive Arbitrary Polynomial Chaos Expansion: A Mini-data-driven Semi-supervised Method for Uncertainty Quantification](https://mlsanity.com/paper/2107.10428/) (2021)
- [PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty](https://mlsanity.com/paper/2202.05063/) (2022)
- [A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas](https://mlsanity.com/paper/1709.08626/) (2017)
- [Consistency regularization-based Deep Polynomial Chaos Neural Network Method for Reliability Analysis](https://mlsanity.com/paper/2203.15655/) (2022)
- [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)
- [Comments on Truncation Errors for Polynomial Chaos Expansions](https://mlsanity.com/paper/1708.07655/) (2017)

## Built on by (most cited first, 4 of 4)

- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [Interpretable epistemic uncertainty decomposition in sequential generative models via polynomial chaos surrogates](https://mlsanity.com/paper/2510.21523/) (2025)
- [A numerical study into neural network surrogate model performance for uncertainty propagation](https://mlsanity.com/paper/2605.16078/) (2026)
- [Uncertainty quantification in mechanics: A unified Bayesian perspective](https://mlsanity.com/paper/2607.18734/) (2026)

## Abstract (arXiv)

Polynomial chaos expansion (PCE) is a classical and widely used surrogate modeling technique in physical simulation and uncertainty quantification. By taking a linear combination of a set of basis polynomials - orthonormal with respect to the distribution of uncertain input parameters - PCE enables tractable inference of key statistical quantities such as (conditional) means, variances, covariances, and Sobol sensitivity indices, which are essential for understanding the modeled system and identifying influential parameters and their interactions. The applicability of PCE to high-dimensional problems is limited by poor scalability, as the number of basis functions grows exponentially with the number of parameters. In this paper, we address this challenge by combining PCE with ideas from tractable probabilistic circuits, resulting in deep polynomial chaos expansion (DeepPCE) - a deep generalization of PCE that scales effectively to high-dimensional input spaces. DeepPCE achieves predictive performance comparable to that of multilayer perceptrons (MLPs), while retaining PCE's ability to compute exact statistical inferences via simple forward passes. In contrast, such computations in MLPs require costly and often inaccurate approximations, such as Monte Carlo integration.