Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

• URL: http://arxiv.org/abs/2206.11324v1
• Date: Wed, 22 Jun 2022 18:57:36 GMT
• Title: Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models
• Authors: Xiao Liu and Xinchao Liu
• Abstract summary: ReducedOrder Models (ROMs) have been widely used to capture the dominant behaviors of high-dimensional systems. A ROM can be obtained, using the well-known Proper Orthogonal Decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes. This paper proposes to use regression trees on Grassmann Manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected.
• Score: 5.738225199806076
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Low dimensional and computationally less expensive Reduced-Order Models (ROMs) have been widely used to capture the dominant behaviors of high-dimensional systems. A ROM can be obtained, using the well-known Proper Orthogonal Decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes which are learned from experimental, simulated or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When a ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann Manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the fact that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters.

Related papers

• Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation [41.13597666007784]
  Dimensionality reduction (DR) offers a useful representation of complex high-dimensional data. Recent DR methods focus on hyperbolic geometry to derive a faithful low-dimensional representation of hierarchical data. This paper presents hGP-LVMs to embed high-dimensional hierarchical data with implicit continuity via nonparametric estimation.
  arXiv  Detail & Related papers  (2024-10-22T05:07:30Z)
• A Statistical Machine Learning Approach for Adapting Reduced-Order Models using Projected Gaussian Process [4.658371840624581]
  Proper Orthogonal Decomposition (POD) computes optimal basis modes that span a low-dimensional subspace where the Reduced-Order Models (ROMs) reside. This paper proposes a Projected Gaussian Process (pGP) and formulates the problem of adapting POD basis as a supervised statistical learning problem. Numerical examples are presented to demonstrate the advantages of the proposed pGP for adapting POD basis against parameter changes.
  arXiv  Detail & Related papers  (2024-10-18T00:02:43Z)
• Diffusion posterior sampling for simulation-based inference in tall data settings [53.17563688225137]
  Simulation-based inference ( SBI) is capable of approximating the posterior distribution that relates input parameters to a given observation. In this work, we consider a tall data extension in which multiple observations are available to better infer the parameters of the model. We compare our method to recently proposed competing approaches on various numerical experiments and demonstrate its superiority in terms of numerical stability and computational cost.
  arXiv  Detail & Related papers  (2024-04-11T09:23:36Z)
• Generative Modeling with Phase Stochastic Bridges [49.4474628881673]
  Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. We introduce a novel generative modeling framework grounded in textbfphase space dynamics Our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.
  arXiv  Detail & Related papers  (2023-10-11T18:38:28Z)
• Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network [59.79008107609297]
  We propose in this paper to approximate the joint posterior over the structure of a Bayesian Network. We use a single GFlowNet whose sampling policy follows a two-phase process. Since the parameters are included in the posterior distribution, this leaves more flexibility for the local probability models.
  arXiv  Detail & Related papers  (2023-05-30T19:16:44Z)
• An iterative multi-fidelity approach for model order reduction of multi-dimensional input parametric PDE systems [0.0]
  We propose a sampling parametric strategy for the reduction of large-scale PDE systems with multidimensional input parametric spaces. It is achieved by exploiting low-fidelity models throughout the parametric space to sample points using an efficient sampling strategy. Since the proposed methodology leverages the use of low-fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage.
  arXiv  Detail & Related papers  (2023-01-23T15:25:58Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• Surrogate-based variational data assimilation for tidal modelling [0.0]
  Data assimilation (DA) is widely used to combine physical knowledge and observations. In a context of climate change, old calibrations can not necessarily be used for new scenarios. This raises the question of DA computational cost. Two methods are proposed to replace the complex model by a surrogate.
  arXiv  Detail & Related papers  (2021-06-08T07:39:38Z)
• 3D Human Pose and Shape Regression with Pyramidal Mesh Alignment Feedback Loop [128.07841893637337]
  Regression-based methods have recently shown promising results in reconstructing human meshes from monocular images. Minor deviation in parameters may lead to noticeable misalignment between the estimated meshes and image evidences. We propose a Pyramidal Mesh Alignment Feedback (PyMAF) loop to leverage a feature pyramid and rectify the predicted parameters.
  arXiv  Detail & Related papers  (2021-03-30T17:07:49Z)
• A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations [0.2148535041822524]
  We develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields.
  arXiv  Detail & Related papers  (2021-03-10T17:01:42Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.