A manifold learning approach to nonlinear model order reduction of quasi-static problems in solid mechanics

Abstract

The proper orthogonal decomposition (POD) -- a popular projection-based model order reduction (MOR) method -- may require significant model dimensionalities to successfully capture a nonlinear solution manifold resulting from a parameterised quasi-static solid-mechanical problem. The local basis method by Amsallem et al. [1] addresses this deficiency by introducing a locally, rather than globally, linear approximation of the solution manifold. However, this generally successful approach comes with some limitations, especially in the data-poor setting. In this proof-of-concept investigation, we instead propose a graph-based manifold learning approach to nonlinear projection-based MOR which uses a global, continuously nonlinear approximation of the solution manifold. Approximations of local tangents to the solution manifold, which are necessary for a Galerkin scheme, are computed in the online phase. As an example application for the resulting nonlinear MOR algorithms, we consider simple representative volume element computations. On this example, the manifold learning approach Pareto-dominates the POD and local basis method in terms of the error and runtime achieved using a range of model dimensionalities.

Figures (22)

• Figure 1: Illustration of a $\delta=1$ dimensional solution manifold $\mathcal{M}_{\bm{u}}$ (black line) in a $D=3$ dimensional solution space, approximated via a $d=2$ dimensional approximation space $\mathcal{M}_{\bm{\bar{u}}}$ (surface with blue-red colour gradient). Snapshots are shown as black dots. Note that here, $\mathcal{M}_{\bm{\bar{u}}}\supset \mathcal{M}_{\bm{u}}$.
• Figure 2: Illustration of dimensionality reduction using the POD. Note that low-dimensional visualisations of manifold learning techniques obscure some of the nuances of dimensionality reduction.
• Figure 3: Illustration of dimensionality reduction using the local basis method. Cluster centroids shown as larger spheres. Note that low-dimensional visualisations of manifold learning techniques obscure some of the nuances of dimensionality reduction.
• Figure 4: Illustration of dimensionality reduction using Laplacian Eigenmaps or Locally Linear Embedding. Note that low-dimensional visualisations of manifold learning techniques obscure some of the nuances of dimensionality reduction.
• Figure 5: Illustration of global linearisation by Pyt:2018:muo: linear approximation of solution manifold based on reduced and original coordinates of neighbouring snapshot vectors $\bm{Y}_n \in \mathbb{R}^{d \times n}$ and $\bm{U}_n \in \mathbb{R}^{D \times n}$. Note that low-dimensional visualisations of manifold learning techniques obscure some of the nuances of dimensionality reduction.