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ML-based identification of the interface regions for coupling local and nonlocal models

Noujoud Nader, Patrick Diehl, Marta D'Elia, Christian Glusa, Serge Prudhomme

TL;DR

This work tackles the challenge of automatically identifying the interface between local and nonlocal regions in local-nonlocal coupling. It develops a CNN-based supervised framework that ingests external loading data and outputs per-node labels (local vs nonlocal), trained against reference VHCM solutions with a fully nonlocal baseline, using a relative-error tolerance to select ground-truth configurations. The authors compare a full-domain input approach with a window-based node-wise approach, finding that windowing yields high accuracy (≈0.96) and F1-scores (≈0.97) while maintaining low coupling errors on one-dimensional tests; this demonstrates a viable path toward automating coupling region identification. The results suggest significant potential to enhance computational efficiency and accuracy in material-science simulations, with future work extending to higher dimensions, variable horizons, and graph-based learning methods.

Abstract

Local-nonlocal coupling approaches combine the computational efficiency of local models and the accuracy of nonlocal models. However, the coupling process is challenging, requiring expertise to identify the interface between local and nonlocal regions. This study introduces a machine learning-based approach to automatically detect the regions in which the local and nonlocal models should be used in a coupling approach. This identification process uses the loading functions and provides as output the selected model at the grid points. Training is based on datasets of loading functions for which reference coupling configurations are computed using accurate coupled solutions, where accuracy is measured in terms of the relative error between the solution to the coupling approach and the solution to the nonlocal model. We study two approaches that differ from one another in terms of the data structure. The first approach, referred to as the full-domain input data approach, inputs the full load vector and outputs a full label vector. In this case, the classification process is carried out globally. The second approach consists of a window-based approach, where loads are preprocessed and partitioned into windows and the problem is formulated as a node-wise classification approach in which the central point of each window is treated individually. The classification problems are solved via deep learning algorithms based on convolutional neural networks. The performance of these approaches is studied on one-dimensional numerical examples using F1-scores and accuracy metrics. In particular, it is shown that the windowing approach provides promising results, achieving an accuracy of 0.96 and an F1-score of 0.97. These results underscore the potential of the approach to automate coupling processes, leading to more accurate and computationally efficient solutions for material science applications.

ML-based identification of the interface regions for coupling local and nonlocal models

TL;DR

This work tackles the challenge of automatically identifying the interface between local and nonlocal regions in local-nonlocal coupling. It develops a CNN-based supervised framework that ingests external loading data and outputs per-node labels (local vs nonlocal), trained against reference VHCM solutions with a fully nonlocal baseline, using a relative-error tolerance to select ground-truth configurations. The authors compare a full-domain input approach with a window-based node-wise approach, finding that windowing yields high accuracy (≈0.96) and F1-scores (≈0.97) while maintaining low coupling errors on one-dimensional tests; this demonstrates a viable path toward automating coupling region identification. The results suggest significant potential to enhance computational efficiency and accuracy in material-science simulations, with future work extending to higher dimensions, variable horizons, and graph-based learning methods.

Abstract

Local-nonlocal coupling approaches combine the computational efficiency of local models and the accuracy of nonlocal models. However, the coupling process is challenging, requiring expertise to identify the interface between local and nonlocal regions. This study introduces a machine learning-based approach to automatically detect the regions in which the local and nonlocal models should be used in a coupling approach. This identification process uses the loading functions and provides as output the selected model at the grid points. Training is based on datasets of loading functions for which reference coupling configurations are computed using accurate coupled solutions, where accuracy is measured in terms of the relative error between the solution to the coupling approach and the solution to the nonlocal model. We study two approaches that differ from one another in terms of the data structure. The first approach, referred to as the full-domain input data approach, inputs the full load vector and outputs a full label vector. In this case, the classification process is carried out globally. The second approach consists of a window-based approach, where loads are preprocessed and partitioned into windows and the problem is formulated as a node-wise classification approach in which the central point of each window is treated individually. The classification problems are solved via deep learning algorithms based on convolutional neural networks. The performance of these approaches is studied on one-dimensional numerical examples using F1-scores and accuracy metrics. In particular, it is shown that the windowing approach provides promising results, achieving an accuracy of 0.96 and an F1-score of 0.97. These results underscore the potential of the approach to automate coupling processes, leading to more accurate and computationally efficient solutions for material science applications.
Paper Structure (26 sections, 14 equations, 13 figures, 3 tables)

This paper contains 26 sections, 14 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Description of the models and coupling approach
  3. Classical linear elasticity model
  4. Linearized bond-based peridynamic model
  5. Variable horizon coupling approach
  6. Discretization
  7. Methodology
  8. Overview
  9. Convolution neural networks
  10. The selected CNN architecture:
  11. Computational resources and model parameters:
  12. Classification evaluation metrics
  13. Data generation
  14. Families of loading functions
  15. Generation of the coupling regions
  16. ...and 11 more sections

Figures (13)

  • Figure 1: Definition of the computational domains for the coupled model. Adapted from author's previous work diehl2022coupling.
  • Figure 2: Example of a variable horizon function $\delta_v(x)$ in one dimension. The circles centered at points $x\in (a,a+\delta)$ are representations of the associated domains $H_\delta(x)$ in terms of $\delta_v(x)$. Adapted from diehl2022coupling.
  • Figure 3: Definition of the grid points and degrees of freedom (represented by $\bullet$ for the degrees of freedom associated with the classical linear elasticity model and by $\circ$ for the degrees of freedom associated with the peridynamic model) in VHCM. Adapted from diehl2022coupling.
  • Figure 4: Overview of the training and prediction: (a) Input Load values used for data generation and as network inputs; (b) Numerical simulations: fully nonlocal solution and coupled solution obtained with the reference local-nonlocal domain splitting; (c) CNN network model; (d) Network output: predicted local-nonlocal splitting configuration. Filled circles $\bullet$ indicate local nodes and empty circles $\circ$ indicate nonlocal nodes.
  • Figure 5: Architecture of the proposed CNN model.
  • ...and 8 more figures