# Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models

arXiv: https://arxiv.org/abs/2107.09814 · HTML: https://mlsanity.com/paper/2107.09814/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 88 references that are in the map.
- First posted on arXiv 21 Jul 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [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)
- [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)
- [Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification](https://mlsanity.com/paper/1812.09444/) (2018)
- [Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning](https://mlsanity.com/paper/1808.02229/) (2018)
- [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)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design](https://mlsanity.com/paper/1712.10131/) (2017)
- [Compressive sensing adaptation for polynomial chaos expansions](https://mlsanity.com/paper/1801.01961/) (2018)
- [Grassmannian diffusion maps based dimension reduction and classification for high-dimensional data](https://mlsanity.com/paper/2009.07547/) (2020)
- [Expanding the Family of Grassmannian Kernels: An Embedding Perspective](https://mlsanity.com/paper/1407.1123/) (2014)
- [Non-intrusive Surrogate Modeling for Parametrized Time-dependent PDEs using Convolutional Autoencoders](https://mlsanity.com/paper/2101.05555/) (2021)
- [Robust Adaptive Least Squares Polynomial Chaos Expansions in High-Frequency Applications](https://mlsanity.com/paper/1912.07725/) (2019)

## Built on by (most cited first, 16 of 16)

- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/2401.16683/) (2024)
- [Grassmannian diffusion maps based surrogate modeling via geometric harmonics](https://mlsanity.com/paper/2109.13805/) (2021)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [Deep Polynomial Chaos Expansion](https://mlsanity.com/paper/2507.21273/) (2025)
- [Data-driven Uncertainty Quantification in Computational Human Head Models](https://mlsanity.com/paper/2110.15553/) (2021)
- [Transient anisotropic kernel for probabilistic learning on manifolds](https://mlsanity.com/paper/2407.21435/) (2024)
- [Dimensionality reduction can be used as a surrogate model for high-dimensional forward uncertainty quantification](https://mlsanity.com/paper/2402.04582/) (2024)
- [Stratospheric aerosol source inversion: Noise, variability, and uncertainty quantification](https://mlsanity.com/paper/2409.06846/) (2024)
- [A Resolution Independent Neural Operator](https://mlsanity.com/paper/2407.13010/) (2024)
- [Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modeling](https://mlsanity.com/paper/2510.22375/) (2025)
- [Structured Neural Chaos: An Adaptive Surrogate Modeling Framework for Functional Uncertainty Quantification and Global Sensitivity Analysis](https://mlsanity.com/paper/2607.28903/) (2026)

## Abstract (arXiv)

In this work we introduce a manifold learning-based method for uncertainty quantification (UQ) in systems describing complex spatiotemporal processes. Our first objective is to identify the embedding of a set of high-dimensional data representing quantities of interest of the computational or analytical model. For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear dimension reduction technique which allows us to reduce the dimensionality of the data and identify meaningful geometric descriptions in a parsimonious and inexpensive manner. Polynomial chaos expansion is then used to construct a mapping between the stochastic input parameters and the diffusion coordinates of the reduced space. An adaptive clustering technique is proposed to identify an optimal number of clusters of points in the latent space. The similarity of points allows us to construct a number of geometric harmonic emulators which are finally utilized as a set of inexpensive pre-trained models to perform an inverse map of realizations of latent features to the ambient space and thus perform accurate out-of-sample predictions. Thus, the proposed method acts as an encoder-decoder system which is able to automatically handle very high-dimensional data while simultaneously operating successfully in the small-data regime. The method is demonstrated on two benchmark problems and on a system of advection-diffusion-reaction equations which model a first-order chemical reaction between two species. In all test cases, the proposed method is able to achieve highly accurate approximations which ultimately lead to the significant acceleration of UQ tasks.