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A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients

Publication ,  Journal Article
Chen, P; Hu, T; Guilleminot, J
Published in: Computer Methods in Applied Mechanics and Engineering
February 1, 2024

Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov n-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN-based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy.

Duke Scholars

Published In

Computer Methods in Applied Mechanics and Engineering

DOI

ISSN

0045-7825

Publication Date

February 1, 2024

Volume

419

Related Subject Headings

  • Applied Mathematics
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 01 Mathematical Sciences
 

Citation

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Chen, P., Hu, T., & Guilleminot, J. (2024). A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients. Computer Methods in Applied Mechanics and Engineering, 419. https://doi.org/10.1016/j.cma.2023.116684
Chen, P., T. Hu, and J. Guilleminot. “A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients.” Computer Methods in Applied Mechanics and Engineering 419 (February 1, 2024). https://doi.org/10.1016/j.cma.2023.116684.
Chen P, Hu T, Guilleminot J. A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients. Computer Methods in Applied Mechanics and Engineering. 2024 Feb 1;419.
Chen, P., et al. “A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients.” Computer Methods in Applied Mechanics and Engineering, vol. 419, Feb. 2024. Scopus, doi:10.1016/j.cma.2023.116684.
Chen P, Hu T, Guilleminot J. A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients. Computer Methods in Applied Mechanics and Engineering. 2024 Feb 1;419.
Journal cover image

Published In

Computer Methods in Applied Mechanics and Engineering

DOI

ISSN

0045-7825

Publication Date

February 1, 2024

Volume

419

Related Subject Headings

  • Applied Mathematics
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 01 Mathematical Sciences