Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification
arXiv:2401.16683 · doi:10.1016/j.jcp.2024.113443
Abstract
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
arXiv comment50 pages, 17 figures
References in corpus (23)
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows
- Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification
- A non-adapted sparse approximation of PDEs with stochastic inputs
- Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media
- Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification
- Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark
- Sparse Pseudospectral Approximation Method
- Enhancing -minimization estimates of polynomial chaos expansions using basis selection
- Least Squares Polynomial Chaos Expansion: A Review of Sampling Strategies
- Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design
- Data-driven probability concentration and sampling on manifold
- Geomstats: A Python Package for Riemannian Geometry in Machine Learning
- Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Network
- Polynomial chaos expansions for dependent random variables
- Compressive sensing adaptation for polynomial chaos expansions
- Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold
- Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC)
- On the influence of over-parameterization in manifold based surrogates and deep neural operators
- Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets
- Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations
- Data-driven Uncertainty Quantification in Computational Human Head Models
- Grassmannian diffusion maps based surrogate modeling via geometric harmonics