Convolutional Neural Network-based Reduced-order Modeling For Parametric Nonlocal PDEs
Abstract
In this paper, we propose a convolutional neural network (CNN) based reduced-order modeling (ROM) to solve parametric nonlocal partial differential equations (PDEs). Our method consists of two main components: dimensional reduction with convolutional autoencoder (CAE) and latent-space modeling with CNN or long short-term memory (LSTM) networks. Our neural network-based ROM bypasses the main challenges faced by intrusive approaches for nonlocal problems, such as non-affine parameter dependence and kernel singularities. To address nonlocal inhomogeneous boundary conditions, we introduce two effective strategies. Additionally, we present two approaches for incorporating parameters into the latent space and demonstrate that CNN mappings are particularly efficient for problems with high-dimensional parameter spaces. Our results provide the evidence that deep CAEs can successfully capture nonlocal behaviors, highlighting the promising potential of neural network-based ROMs for nonlocal PDEs. To the best of our knowledge, our method is the first neural network-based ROM methods developed for nonlocal problems. Extensive numerical experiments, including spatial and temporal nonlocal models, demonstrate that our neural network-based ROMs are effective in solving nonlocal problems. Moreover, our studies show that the compression capability of CAE outperforms traditional projection-based methods, especially when handling complex nonlinear problems.
Recommended Citation
Y. Wang et al., "Convolutional Neural Network-based Reduced-order Modeling For Parametric Nonlocal PDEs," Computer Methods in Applied Mechanics and Engineering, vol. 444, article no. 118084, Elsevier, Sep 2025.
The definitive version is available at https://doi.org/10.1016/j.cma.2025.118084
Department(s)
Mathematics and Statistics
Keywords and Phrases
Convolutional autoencoder; Long-short term memory; Nonlocal inhomogeneous boundary conditions; Nonlocal problems; Parametric model reduction; Reduced order modeling
International Standard Serial Number (ISSN)
0045-7825
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2025 Elsevier, All rights reserved.
Publication Date
01 Sep 2025
Comments
National Science Foundation, Grant DMS-1953177