mlsanityData to 7 Oct 2026

Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification

arXiv 2401.16683 ↗First posted 30 Jan 2024stat.ML cs.LG math.DScited by 6

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Abstract (arXiv)

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemanniann K-means and the minimization of the sample Frechet variance on the Grassmann manifold to identify "local" principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Benard convection problem

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 24 of its 99 references that are in the map. First posted on arXiv 30 Jan 2024.

Builds on

24 of 99 references.

References · 24

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
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  6. 2016Building Deep Networks on Grassmann Manifoldscited by 58
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  12. 2020Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Networkcited by 17
12 more
  1. 2018Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approachcited by 16
  2. 2021Manifold learning-based polynomial chaos expansions for high-dimensional surrogate modelscited by 16
  3. 2023Physics-Informed Polynomial Chaos Expansionscited by 15
  4. 2020Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasetscited by 13
  5. 2020Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applicationscited by 11
  6. 2017Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Designcited by 10
  7. 2023UQpy v4.1: Uncertainty Quantification with Pythoncited by 10
  8. 2018Compressive sensing adaptation for polynomial chaos expansionscited by 9
  9. 2021Grassmannian diffusion maps based surrogate modeling via geometric harmonicscited by 5
  10. 2019Robust Adaptive Least Squares Polynomial Chaos Expansions in High-Frequency Applicationscited by 4
  11. 2021Data-driven Uncertainty Quantification in Computational Human Head Modelscited by 3
  12. 2015Low Rank Representation on Grassmann Manifolds: An Extrinsic Perspectivecited by 2

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Cited by year

Citing papers per year: 2024: 3; 2025: 3.

6 papers.

  1. 2025Neural Chaos: A Spectral Stochastic Neural Operatorcited by 21
  2. 2024Bayesian neural networks for predicting uncertainty in full-field material responsecited by 5
  3. 2024Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approachcited by 2
  4. 2024A Resolution Independent Neural Operatorcited by 0
  5. 2025Learning task-specific predictive models for scientific computingcited by 0
  6. 2025Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modelingcited by 0