mlsanityData to 14 Oct 2026

Deep Polynomial Chaos Expansion

arXiv 2507.21273 ↗First posted 28 Jul 2025cs.LGcited by 4

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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.

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

28 of 97 references.

References · 28

  1. 2017A Unified Approach to Interpreting Model Predictionscited by 6,555
  2. 2020A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challengescited by 819
  3. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  4. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  5. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  6. 2020Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuitscited by 103
  7. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  8. 2019Orthogonal Deep Neural Networkscited by 72
  9. 2020Sparse Polynomial Chaos Expansions: Literature Survey and Benchmarkcited by 44
  10. 2022Scaling Up Probabilistic Circuits by Latent Variable Distillationcited by 35
  11. 2021Adaptive wavelet distillation from neural networks through interpretationscited by 32
  12. 2018Data-driven polynomial chaos expansion for machine learning regressioncited by 25
16 more
  1. 2024What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?cited by 22
  2. 2015Polynomial-Chaos-based Krigingcited by 21
  3. 2016Visualizing and Understanding Sum-Product Networkscited by 18
  4. 2021Manifold learning-based polynomial chaos expansions for high-dimensional surrogate modelscited by 16
  5. 2021Existence, Stability and Scalability of Orthogonal Convolutional Neural Networkscited by 16
  6. 2020Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applicationscited by 11
  7. 2024A Guide to Feature Importance Methods for Scientific Inferencecited by 11
  8. 2021Bayesian Surrogate Analysis and Uncertainty Propagationcited by 10
  9. 2021PCE-PINNs: Physics-Informed Neural Networks for Uncertainty Propagation in Ocean Modelingcited by 9
  10. 2023Bayesian Structure Scores for Probabilistic Circuitscited by 7
  11. 2021Deep Adaptive Arbitrary Polynomial Chaos Expansion: A Mini-data-driven Semi-supervised Method for Uncertainty Quantificationcited by 6
  12. 2022PCENet: High Dimensional Surrogate Modeling for Learning Uncertaintycited by 5
  13. 2017A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulascited by 3
  14. 2022Consistency regularization-based Deep Polynomial Chaos Neural Network Method for Reliability Analysiscited by 3
  15. 2023The Deep Arbitrary Polynomial Chaos Neural Network or how Deep Artificial Neural Networks could benefit from Data-Driven Homogeneous Chaos Theorycited by 3
  16. 2017Comments on Truncation Errors for Polynomial Chaos Expansionscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 2; 2026: 2.

4 papers.

  1. 2025How to Square Tensor Networks and Circuits Without Squaring Themcited by 5
  2. 2025Interpretable epistemic uncertainty decomposition in sequential generative models via polynomial chaos surrogatescited by 0
  3. 2026A numerical study into neural network surrogate model performance for uncertainty propagationcited by 0
  4. 2026Uncertainty quantification in mechanics: A unified Bayesian perspectivecited by 0