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

arXiv: https://arxiv.org/abs/2401.16683 · HTML: https://mlsanity.com/paper/2401.16683/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 99 references that are in the map.
- First posted on arXiv 30 Jan 2024.

## Builds on (most cited first, 24 of 24)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification](https://mlsanity.com/paper/1812.09444/) (2018)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Network](https://mlsanity.com/paper/2004.13560/) (2020)
- [Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approach](https://mlsanity.com/paper/1812.06309/) (2018)
- [Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models](https://mlsanity.com/paper/2107.09814/) (2021)
- [Physics-Informed Polynomial Chaos Expansions](https://mlsanity.com/paper/2309.01697/) (2023)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications](https://mlsanity.com/paper/2009.04800/) (2020)
- [Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design](https://mlsanity.com/paper/1712.10131/) (2017)
- [UQpy v4.1: Uncertainty Quantification with Python](https://mlsanity.com/paper/2305.09572/) (2023)
- [Compressive sensing adaptation for polynomial chaos expansions](https://mlsanity.com/paper/1801.01961/) (2018)
- [Grassmannian diffusion maps based surrogate modeling via geometric harmonics](https://mlsanity.com/paper/2109.13805/) (2021)
- [Robust Adaptive Least Squares Polynomial Chaos Expansions in High-Frequency Applications](https://mlsanity.com/paper/1912.07725/) (2019)
- [Data-driven Uncertainty Quantification in Computational Human Head Models](https://mlsanity.com/paper/2110.15553/) (2021)
- [Low Rank Representation on Grassmann Manifolds: An Extrinsic Perspective](https://mlsanity.com/paper/1504.01807/) (2015)

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

- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [A Resolution Independent Neural Operator](https://mlsanity.com/paper/2407.13010/) (2024)
- [Learning task-specific predictive models for scientific computing](https://mlsanity.com/paper/2506.03835/) (2025)
- [Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modeling](https://mlsanity.com/paper/2510.22375/) (2025)

## 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