RETRACTED ARTICLE: Deep Learning Aided Surrogate Model for Real Prediction of Diffusion Process

Articles in Press

Document Type : Research Paper

Authors
Wenda Yan 1
Hongzhi Wang 2
Shuisheng Zhang 1
Jian Zhang 1
1 College of Science, Qiqihar University, Qiqihar, 110500, China
2 College of Architecture and Civil Engineering, Qiqihar, 110500, China
10.22055/jacm.2025.48691.5429
Abstract
This study presents a hybrid deep learning model combining U-Net and LSTM architectures to accurately predict the spatiotemporal diffusion processes while maintaining close accuracy. A comprehensive dataset of 200 FEM simulations is developed, and each of them captures 20 time-steps of 2D diffusion under randomized initial conditions with Dirichlet boundaries. The model's key kernel is its integration of U-Net's spatial feature extraction with LSTM's temporal modeling, enabling efficient learning of diffusion dynamics. Using just five input frames, it achieves high accuracy while reducing computation time by over 90% compared to FEM simulation. Extensive validation confirms strong generalization across varying initial conditions and geometries. While currently applied to 2D isotropic diffusion with fixed boundaries, the architecture is adaptable for future extensions, including 3D domains, anisotropic diffusion, and dynamic boundary conditions. This work offers both a computationally efficient alternative to FEM and a generalizable navigation for PDE-driven physical systems with more complex physical constraints and more complex scenarios.
Keywords
Deep learning
Neural network model
Diffusion process
Surrogate model
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

⊗ This article was retracted on 11 September 2026

 

[1] Jing, L., Tian, Y., Self-Supervised Visual Feature Learning with Deep Neural Networks: A Survey, IEEE Transactions on Pattern Analysis and Machine Intelligence, 43, 2019, 4037-58.
[2] Schirrmeister, R., Gemein, L., Eggensperger, K., et al., Deep learning with convolutional neural networks for decoding and visualization of EEG pathology, Proceedings of the 2017 IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2017.
[3] Panayides, A.S., Amini, A., Filipovic, N.D., et al., AI in Medical Imaging Informatics: Current Challenges and Future Directions, IEEE Journal of Biomedical and Health Informatics, 24(7), 2020, 1837-57.
[4] Khalil, R.A., Jones, E., Babar, M.I., et al., Speech Emotion Recognition Using Deep Learning Techniques: A Review, IEEE Access, 7, 2019, 117327-45.
[5] Nassif, A.B., Shahin, I., Attili, I., et al., Speech Recognition Using Deep Neural Networks: A Systematic Review, IEEE Access, 7, 2019, 19143-65.
[6] Noda, K., Yamaguchi, Y., Nakadai, K., et al., Audio-visual speech recognition using deep learning, Applied Intelligence, 42(4), 2015, 722-37.
[7] Otter, D.W., Medina, J.R., Kalita, J.K., A Survey of the Usages of Deep Learning for Natural Language Processing, IEEE Transactions on Neural Networks and Learning Systems, 32(2), 2021, 604-24.
[8] Zhang, L., Wang, S., Liu, B., Deep learning for sentiment analysis: A survey, WIREs Data Mining and Knowledge Discovery, 8(4), 2018, e1253.
[9] Young, T., Hazarika, D., Poria, S., et al., Recent Trends in Deep Learning Based Natural Language Processing [Review Article], IEEE Computational Intelligence Magazine, 13(3), 2018, 55-75.
[10] Raissi, M., Perdikaris, P., Karniadakis, G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, 378, 2019, 686-707.
[11] Karniadakis, G.E., Kevrekidis, I.G., Lu, L., et al., Physics-informed machine learning, Nature Reviews Physics, 3(6), 2021, 422-40.
[12] Raissi, M., Karniadakis, G.E., Hidden physics models: Machine learning of nonlinear partial differential equations, Journal of Computational Physics, 357, 2018, 125-41.
[13] Lu, L., Meng, X., Mao, Z., et al., DeepXDE: A Deep Learning Library for Solving Differential Equations, SIAM Review, 63(1), 2021, 208-28.
[14] Wang, S., Karniadakis, G.E., GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domains, Computer Methods in Applied Mechanics and Engineering, 429, 2024, 117189.
[15] Cuomo, S., Di Cola, V.S., Giampaolo, F., et al., Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next, Journal of Scientific Computing, 92(3), 2022, 88.
[16] Yuan, L., Ni, Y.-Q., Deng, X.-Y., et al., A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations, Journal of Computational Physics, 462, 2022, 111260.
[17] Meng, Z., Qian, Q., Xu, M., et al., PINN-FORM: A new physics-informed neural network for reliability analysis with partial differential equation, Computer Methods in Applied Mechanics and Engineering, 414, 2023, 116172.
[18] Cheng, S., Chen, J., Anastasiou, C., et al., Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models, Journal of Scientific Computing, 94(1), 2022, 11.
[19] Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M., Automatic differentiation in machine learning: a survey, Journal of Machine Learning Research, 18(153), 2018, 1-43.
[20] Yao, Y., Guo, J., Gu, T., A deep learning method for multi-material diffusion problems based on physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering, 417, 2023, 116395.
[21] Hsu, Y.-C., Yu, C.-H., Buehler, M.J., Using Deep Learning to Predict Fracture Patterns in Crystalline Solids, Matter, 3(1), 2020, 197-211.
[22] Wei, H., Yao, H., Pang, Y., et al., Fracture pattern prediction with random microstructure using a physics-informed deep neural networks, Engineering Fracture Mechanics, 268, 2022, 108497.
[23] Worthington, M., Chew, H.B., Crack path predictions in heterogeneous media by machine learning, Journal of the Mechanics and Physics of Solids, 172, 2023, 105188.
[24] Buehler, E.L., Buehler, M.J., End-to-end prediction of multimaterial stress fields and fracture patterns using cycle-consistent adversarial and transformer neural networks, Biomedical Engineering Advances, 4, 2022, 100038.
[25] Sepasdar, R., Karpatne, A., Shakiba, M., A data-driven approach to full-field nonlinear stress distribution and failure pattern prediction in composites using deep learning, Computer Methods in Applied Mechanics and Engineering, 397, 2022, 115126.
[26] Guilleminot, J., Dolbow, J.E., Data-driven enhancement of fracture paths in random composites, Mechanics Research Communications, 103, 2020, 103443.
[27] Zhuang, Y., Cheng, S., Kovalchuk, N., et al., Ensemble latent assimilation with deep learning surrogate model: application to drop interaction in a microfluidics device, Lab on a Chip, 22(17), 2022, 3187-202.
[28] Wang, H., Zhou, H., Cheng, S., Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint, Computer Methods in Applied Mechanics and Engineering, 43, 22024, 117339.
[29] Najafi Koopas, R., Rezaei, S., Rauter, N., et al., Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space, Computational Mechanics, 75(4), 2025, 1377-406.
[30] Hu, C., Martin, S., Dingreville, R., Accelerating phase-field predictions via recurrent neural networks learning the microstructure evolution in latent space, Computer Methods in Applied Mechanics and Engineering, 397, 2022, 115128.
[31] Montes de Oca Zapiain, D., Stewart, J.A., Dingreville, R., Accelerating phase-field-based microstructure evolution predictions via surrogate models trained by machine learning methods, npj Computational Materials, 7(1), 2021, 3.
[32] Yabansu, Y.C., Steinmetz, P., Hötzer, J., et al., Extraction of reduced-order process-structure linkages from phase-field simulations, Acta Materialia, 124, 2017, 182-94.
[33] Oommen, V., Shukla, K., Goswami, S., et al., Learning two-phase microstructure evolution using neural operators and autoencoder architectures, npj Computational Materials, 8(1), 2022, 190.
[34] Zhou, H., Cheng, S., Arcucci, R., Multi-fidelity physics constrained neural networks for dynamical systems, Computer Methods in Applied Mechanics and Engineering, 420, 2024, 116758.
[35] Zhou, H., Cheng, S., Improving long-term autoregressive spatiotemporal predictions: A proof of concept with fluid dynamics, Computer Methods in Applied Mechanics and Engineering, 447, 2025, 118332.
[36] Cheng, S., Zhuang, Y., Kahouadji, L., et al., Multi-domain encoder–decoder neural networks for latent data assimilation in dynamical systems, Computer Methods in Applied Mechanics and Engineering, 430, 2024, 117201.
[37] Ronneberger, O., Fischer, P., Brox, T., U-net: Convolutional networks for biomedical image segmentation, International Conference on Medical Image Computing and Computer-Assisted Intervention, 2015.
[38] Graves, A., Mohamed, A.R., Hinton, G., Speech recognition with deep recurrent neural networks, Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, 2013.
[39] Palangi, H., Deng, L., Shen, Y., et al., Deep Sentence Embedding Using Long Short-Term Memory Networks: Analysis and Application to Information Retrieval, IEEE/ACM Transactions on Audio, Speech, and Language Processing, 24(4), 2016, 694-707.
[40] Hewamalage, H., Bergmeir, C., Bandara, K., Recurrent Neural Networks for Time Series Forecasting: Current status and future directions, International Journal of Forecasting, 37(1), 2021, 388-427.
[41] Hochreiter, S., Schmidhuber, J., Long Short-Term Memory, Neural Computation, 9(8), 1997, 1735-80.
[42] Yu, Y., Si, X., Hu, C., et al., A Review of Recurrent Neural Networks: LSTM Cells and Network Architectures, Neural Computation, 31(7), 2019, 1235-1270.
[43] Hua, Y., Zhao, Z., Li, R., et al., Deep Learning with Long Short-Term Memory for Time Series Prediction, IEEE Communications Magazine, 57(6), 2019, 114-119.
Journal of Applied and Computational Mechanics

Articles in Press, Corrected Proof
Available Online from 14 November 2025