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Volume 17, Issue 4
Physics-Informed Deep Learning Method for Surrogate Modeling in Elastic Mechanics

Dan Li, Xinming Lu, Hongjuan Wang, Guibin Liu & Xin Liu

Adv. Appl. Math. Mech., 17 (2025), pp. 1239-1258.

Published online: 2025-05

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  • Abstract

In this paper, we utilize Physics-Informed Neural Networks (PINNs) without any labeled data to solve the fourth-order partial differential equation governing the bending of thin plates. We meticulously formulate and apply our framework to the bending problem of elastic thin plates. Our findings indicate that defining each solution variable through an independent network, meaning without shared network parameters, yields superior performance compared to a single-network model with multiple outputs. During the training process, we discovered that more accurate results were achieved by adjusting the network architecture to strictly satisfy the boundary conditions, rather than incorporating them as part of the loss function. Remarkably, the PINNs are capable of obtaining relatively good results without the use of any labeled data during training, irrespective of whether soft or hard constraints are applied. In the end, the neural network’s predictions for plate deflection, stress, strain, bending, and shear force all have errors less than 5% compared to the analytical solutions. In the algorithm implementation phase, we leverage the Python library DeepXDE, which facilitates the training of PINNs by providing an efficient and expedited process.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-17-1239, author = {Li , DanLu , XinmingWang , HongjuanLiu , Guibin and Liu , Xin}, title = {Physics-Informed Deep Learning Method for Surrogate Modeling in Elastic Mechanics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {17}, number = {4}, pages = {1239--1258}, abstract = {

In this paper, we utilize Physics-Informed Neural Networks (PINNs) without any labeled data to solve the fourth-order partial differential equation governing the bending of thin plates. We meticulously formulate and apply our framework to the bending problem of elastic thin plates. Our findings indicate that defining each solution variable through an independent network, meaning without shared network parameters, yields superior performance compared to a single-network model with multiple outputs. During the training process, we discovered that more accurate results were achieved by adjusting the network architecture to strictly satisfy the boundary conditions, rather than incorporating them as part of the loss function. Remarkably, the PINNs are capable of obtaining relatively good results without the use of any labeled data during training, irrespective of whether soft or hard constraints are applied. In the end, the neural network’s predictions for plate deflection, stress, strain, bending, and shear force all have errors less than 5% compared to the analytical solutions. In the algorithm implementation phase, we leverage the Python library DeepXDE, which facilitates the training of PINNs by providing an efficient and expedited process.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0253}, url = {http://global-sci.org/intro/article_detail/aamm/24060.html} }
TY - JOUR T1 - Physics-Informed Deep Learning Method for Surrogate Modeling in Elastic Mechanics AU - Li , Dan AU - Lu , Xinming AU - Wang , Hongjuan AU - Liu , Guibin AU - Liu , Xin JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 1239 EP - 1258 PY - 2025 DA - 2025/05 SN - 17 DO - http://doi.org/10.4208/aamm.OA-2023-0253 UR - https://global-sci.org/intro/article_detail/aamm/24060.html KW - PINNs, elastic thin plates, independent network without labeled data, DeepXDE. AB -

In this paper, we utilize Physics-Informed Neural Networks (PINNs) without any labeled data to solve the fourth-order partial differential equation governing the bending of thin plates. We meticulously formulate and apply our framework to the bending problem of elastic thin plates. Our findings indicate that defining each solution variable through an independent network, meaning without shared network parameters, yields superior performance compared to a single-network model with multiple outputs. During the training process, we discovered that more accurate results were achieved by adjusting the network architecture to strictly satisfy the boundary conditions, rather than incorporating them as part of the loss function. Remarkably, the PINNs are capable of obtaining relatively good results without the use of any labeled data during training, irrespective of whether soft or hard constraints are applied. In the end, the neural network’s predictions for plate deflection, stress, strain, bending, and shear force all have errors less than 5% compared to the analytical solutions. In the algorithm implementation phase, we leverage the Python library DeepXDE, which facilitates the training of PINNs by providing an efficient and expedited process.

Li , DanLu , XinmingWang , HongjuanLiu , Guibin and Liu , Xin. (2025). Physics-Informed Deep Learning Method for Surrogate Modeling in Elastic Mechanics. Advances in Applied Mathematics and Mechanics. 17 (4). 1239-1258. doi:10.4208/aamm.OA-2023-0253
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