Abstract Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \ x_i  x_j \^2} {\sigma ^2} ) $ is widely used in graphbased geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $\sigma $, and a common practice called selftuned kernel adaptively sets a $\sigma _i$ at each point $x_i$ by the $k$nearest neighbor (kNN) distance. When $x_i$s are sampled from a $d$dimensional manifold embedded in a possibly highdimensional space, unlike with fixedbandwidth kernels, theoretical results of graph Laplacian convergence with selftuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted)Laplacian for a new family of kNN selftuned kernels $W^{(\alpha )}_{ij} = k_0( \frac{ \ x_i  x_j \^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $, where $\hat{\rho }$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $\alpha $. When $\alpha = 1$, the limiting operator is the weighted manifold Laplacian $\varDelta _p$. Specifically, we prove the pointwise convergence of $L_N f $ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $\hat{\rho }$ which bounds the relative estimation error $\hat{\rho }  \bar{\rho }/\bar{\rho }$ uniformly with high probability, where $\bar{\rho } = p^{1/d}$ and $p$ is the data density function. Our theoretical results reveal the advantage of the selftuned kernel over the fixedbandwidth kernel via smaller variance error in lowdensity regions. In the algorithm, no prior knowledge of $d$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and handwritten digit image data.
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This content will become publicly available on October 1, 2024
Geometric learning for computational mechanics, Part III: Physicsconstrained response surface of geometrically nonlinear shells
This paper presents a graphmanifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a lowdimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.
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 Award ID(s):
 1846875
 NSFPAR ID:
 10487105
 Editor(s):
 De Lorenzis, Laura; Papadrakakis, Manolis; Zohdi, Tarek I.
 Publisher / Repository:
 Computer Methods in Applied Mechanics and Engineering
 Date Published:
 Journal Name:
 Computer Methods in Applied Mechanics and Engineering
 Volume:
 415
 Issue:
 C
 ISSN:
 00457825
 Page Range / eLocation ID:
 116219
 Subject(s) / Keyword(s):
 ["Machine learning","Graph neural network","Shell","Reduced order modeling"]
 Format(s):
 Medium: X Size: 6.3MB Other: pdf
 Size(s):
 ["6.3MB"]
 Sponsoring Org:
 National Science Foundation
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