基于物理信息深度学习的高分子流变学研究：挑战、方法与应用

祁纪浩; 陈欣; 庞志威; 林增; 王超元; 刘虎; 沙金; 白志山

doi:10.11777/j.issn1000-3304.2025.25155

您当前的位置：

首页 >

文章列表页 >

基于物理信息深度学习的高分子流变学研究：挑战、方法与应用

综述 | 更新时间：2025-09-22

- 基于物理信息深度学习的高分子流变学研究：挑战、方法与应用
- Physics-Informed Deep Learning for Polymer Rheology: Investigating Challenges, Methodologies, and Applications
- 高分子学报 2025年页码：1-16
- 作者机构：
  
  华东理工大学机械与动力工程学院上海 200237
- 作者简介：
  
  [ "沙金，男，1982年生. 华东理工大学副教授、硕士生导师. 2011~2013年在美国威斯康辛麦迪逊大学访学，2014年在华东理工大学获得工学博士学位. 获得上海市科技进步奖二等奖. 主要研究方向为高性能材料成型制造及智能装备，及其在多元组分、功能化表界面微图案及生物芯片系统中的应用. 于Lab on Chip、Biomacromolecules、Biomaterials、Composite Part A等国际知名刊物上发表论文二十余篇." ]
- 基金信息：
  
  国家杰出青年科学基金(基金号 22225804);国家自然科学基金(基金号 22408101);上海市自然科学基金(基金号 25ZR1401085);贵州省科技支撑计划(项目号黔科合支撑 [2025] 一般091)
- DOI：10.11777/j.issn1000-3304.2025.25155
  中图分类号：
- CSTR：32057.14.GFZXB.2025.7457
- 收稿日期：2025-06-26，
  
  录用日期：2025-08-15，
  
  网络出版日期：2025-09-22，
- 稿件说明：
移动端阅览
祁纪浩, 陈欣, 庞志威, 林增, 王超元, 刘虎, 沙金, 白志山. 基于物理信息深度学习的高分子流变学研究：挑战、方法与应用. 高分子学报, doi: 10.11777/j.issn1000-3304.2025.25155

Qi J. H.; Chen X.; Pang Z. W.; Lin Z.; Wang C. Y.; Liu H.; Sha J.; Bai Z. S. Physics-informed deep learning for polymer rheology: investigating challenges, methodologies, and applications. Acta Polymerica Sinica, doi: 10.11777/j.issn1000-3304.2025.25155
祁纪浩, 陈欣, 庞志威, 林增, 王超元, 刘虎, 沙金, 白志山. 基于物理信息深度学习的高分子流变学研究：挑战、方法与应用. 高分子学报, doi: 10.11777/j.issn1000-3304.2025.25155 DOI： CSTR： 32057.14.GFZXB.2025.7457.

Qi J. H.; Chen X.; Pang Z. W.; Lin Z.; Wang C. Y.; Liu H.; Sha J.; Bai Z. S. Physics-informed deep learning for polymer rheology: investigating challenges, methodologies, and applications. Acta Polymerica Sinica, doi: 10.11777/j.issn1000-3304.2025.25155 DOI： CSTR： 32057.14.GFZXB.2025.7457.

摘要

高分子流变学旨在理解材料从微观到宏观的流动与变形特性，但传统建模方法在处理复杂非线性、多尺度模拟及数据解析方面长期面临挑战. 物理信息深度学习(PIDL)通过将流变学物理定律嵌入深度神经网络，减少对大量数据的依赖，提高模型泛化能力和物理合理性，为解决高分子流变学面临的挑战提供新的途径. 通过探讨PIDL的基本原理，包括其模型架构、物理信息损失函数设计(如MSE、MAE及稀疏/残差损失)和损失函数优化方法(如梯度下降、自适应权重调整)，并分析PIDL在高分子流变学中的具体应用，涵盖了概率模型、生成式模型、神经算子、图神经网络和强化学习等多种模型架构，展示其在参数优化、流变特性预测、逆问题求解及数字孪生集成等方面的潜力. 尽管PIDL展现出显著优势，但仍面临数据获取成本高、模型可解释性弱、多尺度求解精度不足及鲁棒性等问题. 未来发展方向包括利用多保真技术、物理信息增强数据、优化算法、结合数字孪生、推断因果关系以及应用元学习和可解释人工智能(XAI)来提升模型性能和实用性.

Abstract

Polymer rheology aims to understand the multiscale flow and deformation of macromolecular materials

yet conventional constitutive approaches have long struggled with pronounced nonlinearities

cross-scale coupling and sparse

noisy data inversion. Physics-informed deep learning (PIDL) embeds rheological conservation laws and constitutive relations directly into deep neural networks

substantially reducing the demand for large labelled datasets while enhancing model generalization and physical interpretability. This review systematically outlines the foundational principles of PIDL

covering network architectures

physics-informed loss functions such as MSE

MAE

sparse and residual variants

and optimization strategies including gradient descent and adaptive weighting. We further survey five PIDL paradigms—probabilistic models

generative models

neural operators

graph neural networks and reinforcement learning—and demonstrate their potential for parameter identification

rheological property prediction

inverse problem solving and digital-twin integration. Despite these advantages

PIDL still contends with high data acquisition costs

limited interpretability

insufficient accuracy across scales and uncertain robustness under extreme conditions. Future research should exploit multi-fidelity techniques

physics-augmented data generation

advanced optimization

causal inference and explainable AI to deliver trustworthy

industrially viable rheological models.

关键词

Keywords

references

Wilson, D. I. What is rheology? Eye , 2018 , 32 ( 2 ), 179 - 183 . doi: 10.1038/eye.2017.267 http://dx.doi.org/10.1038/eye.2017.267

Deshpande A. P. ; Krishnan J. M. ; Sunil Kumar P. B. , ed . Rheology of complex fluids . New York : Springer , 2010 : 171 - 191 . doi: 10.1007/978-1-4419-6494-6 http://dx.doi.org/10.1007/978-1-4419-6494-6

Polychronopoulos N. D. ; Vlachopoulos J. Polymer processing and rheology . Functional Polymers. Cham : Springer International Publishing , 2019 , 133 - 180 . doi: 10.1007/978-3-319-95987-0_4 http://dx.doi.org/10.1007/978-3-319-95987-0_4

Mahesh B. Machine learning algorithms—a review . Int. J. Sci. Res. , 2020 , 9 ( 1 ), 381 - 386 . doi: 10.21275/art20203995 http://dx.doi.org/10.21275/art20203995

Bertolini M. ; Mezzogori D. ; Neroni M. ; Zammori F. Machine Learning for industrial applications: a comprehensive literature review . Expert Syst. Appl. , 2021 , 175 , 114820 . doi: 10.1016/j.eswa.2021.114820 http://dx.doi.org/10.1016/j.eswa.2021.114820

Meng C. Z. ; Griesemer S. ; Cao D. F. ; Seo S. ; Liu Y. When physics meets machine learning: a survey of physics-informed machine learning . Mach. Learn. Comput. Sci. Eng. , 2025 , 1 ( 1 ), 20 . doi: 10.1007/s44379-025-00016-0 http://dx.doi.org/10.1007/s44379-025-00016-0

Hao Z. K. ; Liu S. M. ; Zhang Y. C. ; Ying C. Y. ; Feng Y. ; Su H. ; Zhu J. Physics-informed machine learning: a survey on problems, methods and applications . arXiv E Prints , 2022 , arXiv, 2211.08064.

Karniadakis G. E. ; Kevrekidis I. G. ; Lu L. ; Perdikaris P. ; Wang S. F. ; Yang L. Physics-informed machine learning . Nat. Rev. Phys. , 2021 , 3 ( 6 ), 422 - 440 . doi: 10.1038/s42254-021-00314-5 http://dx.doi.org/10.1038/s42254-021-00314-5

Sharma P. ; Chung W. T. ; Akoush B. ; Ihme M. A review of physics-informed machine learning in fluid mechanics . Energies , 2023 , 16 ( 5 ), 2343 . doi: 10.3390/en16052343 http://dx.doi.org/10.3390/en16052343

Yao L. J. ; Wang J. X. ; Chuai M. Y. ; Lomov S. V. ; Carvelli V. Physics-informed machine learning for loading history dependent fatigue delamination of composite laminates . Compos. Part A Appl. Sci. Manuf. , 2024 , 187 , 108474 . doi: 10.1016/j.compositesa.2024.108474 http://dx.doi.org/10.1016/j.compositesa.2024.108474

Latrach A. ; Malki M. L. ; Morales M. ; Mehana M. ; Rabiei M. A critical review of physics-informed machine learning applications in subsurface energy systems . Geoenergy Sci. Eng. , 2024 , 239 , 212938 . doi: 10.1016/j.geoen.2024.212938 http://dx.doi.org/10.1016/j.geoen.2024.212938

Rao C. P. ; Sun H. ; Liu Y. Physics-informed deep learning for incompressible laminar flows . Theor. Appl. Mech. Lett. , 2020 , 10 ( 3 ), 207 - 212 . doi: 10.1016/j.taml.2020.01.039 http://dx.doi.org/10.1016/j.taml.2020.01.039

Xu G. J. ; Ji C. J. ; Xu Y. ; Yu E. B. ; Cao Z. Y. ; Wu Q. H. ; Lin P. Z. ; Wang J. S. Machine learning in coastal bridge hydrodynamics: a state-of-the-art review . Appl. Ocean. Res. , 2023 , 134 , 103511 . doi: 10.1016/j.apor.2023.103511 http://dx.doi.org/10.1016/j.apor.2023.103511

Zhao F. ; Zhang Z. Y. ; Yang J. J. ; Yu J. X. ; Feng J. Y. ; Zhou H. X. ; Shi C. J. ; Meng F. N. Advanced nonlinear rheology magnetorheological finishing: a review . Chin. J. Aeronaut. , 2024 , 37 ( 4 ), 54 - 92 . doi: 10.1016/j.cja.2023.06.006 http://dx.doi.org/10.1016/j.cja.2023.06.006

Alkalbani A. M. ; Chala G. T. A comprehensive review of nanotechnology applications in oil and gas well drilling operations . Energies , 2024 , 17 ( 4 ), 798 . doi: 10.3390/en17040798 http://dx.doi.org/10.3390/en17040798

Pipintakos G. ; Sreeram A. ; Mirwald J. ; Bhasin A. Engineering bitumen for future asphalt pavements: a review of chemistry, structure and rheology . Mater. Des. , 2024 , 244 , 113157 . doi: 10.1016/j.matdes.2024.113157 http://dx.doi.org/10.1016/j.matdes.2024.113157

Raihan M. K. ; Jagdale P. P. ; Wu S. ; Shao X. C. ; Bostwick J. B. ; Pan X. X. ; Xuan X. C. Flow of non-Newtonian fluids in a single-cavity microchannel . Micromachines , 2021 , 12 ( 7 ), 836 . doi: 10.3390/mi12070836 http://dx.doi.org/10.3390/mi12070836

Moradpour N. ; Yang J. Y. ; Tsai P. A. Liquid foam: fundamentals, rheology, and applications of foam displacement in porous structures . Curr. Opin. Colloid Interface Sci. , 2024 , 74 , 101845 . doi: 10.1016/j.cocis.2024.101845 http://dx.doi.org/10.1016/j.cocis.2024.101845

Ahuja A. ; Lee R. ; Joshi Y. M. Advances and challenges in the high-pressure rheology of complex fluids . Adv. Colloid Interface Sci. , 2021 , 294 , 102472 . doi: 10.1016/j.cis.2021.102472 http://dx.doi.org/10.1016/j.cis.2021.102472

Taira K. ; Hemati M. S. ; Brunton S. L. ; Sun Y. Y. ; Duraisamy K. ; Bagheri S. ; Dawson S. T. M. ; Yeh C. A. Modal analysis of fluid flows: applications and outlook . AIAA j. , 2019 , 58 ( 3 ), 998 - 1022 . doi: 10.2514/1.j058462 http://dx.doi.org/10.2514/1.j058462

Wu S. L. ; Chen Q. Advances and new opportunities in the rheology of physically and chemically reversible polymers . Macromolecules , 2022 , 55 ( 3 ), 697 - 714 . doi: 10.1021/acs.macromol.1c01605 http://dx.doi.org/10.1021/acs.macromol.1c01605

Huang Q. When polymer chains are highly aligned: a perspective on extensional rheology . Macromolecules , 2022 , 55 ( 3 ), 715 - 727 . doi: 10.1021/acs.macromol.1c02262 http://dx.doi.org/10.1021/acs.macromol.1c02262

Münstedt H. Rheological measurements and structural analysis of polymeric materials . Polymers , 2021 , 13 ( 7 ), 1123 . doi: 10.3390/polym13071123 http://dx.doi.org/10.3390/polym13071123

Romberg S. K. ; Kotula A. P. Simultaneous rheology and cure kinetics dictate thermal post-curing of thermoset composite resins for material extrusion . Addit. Manuf. , 2023 , 71 , 103589 . doi: 10.1016/j.addma.2023.103589 http://dx.doi.org/10.1016/j.addma.2023.103589

Ackermann A. C. ; Carosella S. ; Rettenmayr M. ; Fox B. L. ; Middendorf P. Rheology, dispersion, and cure kinetics of epoxy filled with amine- and non-functionalized reduced graphene oxide for composite manufacturing . J. Appl. Polym. Sci. , 2022 , 139 ( 8 ), 51664 . doi: 10.1002/app.51664 http://dx.doi.org/10.1002/app.51664

Song H. W. ; Li X. L. An overview on the rheology, mechanical properties , durability , 3 printingD, and microstructural performance of nanomaterials in cementitious composites. Materials, 2021, 14 ( 11 ), 2950 . doi: 10.3390/ma14112950 http://dx.doi.org/10.3390/ma14112950

Gonzalez-Saiz E. ; Garcia-Gonzalez D. Model-driven identification framework for optimal constitutive modeling from kinematics and rheological arrangement . Comput. Meth. Appl. Mech. Eng. , 2023 , 415 , 116211 . doi: 10.1016/j.cma.2023.116211 http://dx.doi.org/10.1016/j.cma.2023.116211

Larson R. G. ; Desai P. S. Modeling the rheology of polymer melts and solutions . Annu. Rev. Fluid Mech. , 2015 , 47 , 47 - 65 . doi: 10.1146/annurev-fluid-010814-014612 http://dx.doi.org/10.1146/annurev-fluid-010814-014612

Sato T. ; Yoshimoto K. Recent developments on multiscale simulations for rheology and complex flow of polymers . Korea Aust. Rheol. J. , 2024 , 36 ( 4 ), 253 - 269 . doi: 10.1007/s13367-024-00112-2 http://dx.doi.org/10.1007/s13367-024-00112-2

Liu N. P. ; Zhang D. ; Gao H. ; Hu Y. L. ; Duan L. C. Real-time measurement of drilling fluid rheological properties: a review . Sensors , 2021 , 21 ( 11 ), 3592 . doi: 10.3390/s21113592 http://dx.doi.org/10.3390/s21113592

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 . J. Comput. Phys. , 2019 , 378 , 686 - 707 . doi: 10.1016/j.jcp.2018.10.045 http://dx.doi.org/10.1016/j.jcp.2018.10.045

Lai Z. L. ; Mylonas C. ; Nagarajaiah S. ; Chatzi E. Structural identification with physics-informed neural ordinary differential equations . J. Sound Vib. , 2021 , 508 , 116196 . doi: 10.1016/j.jsv.2021.116196 http://dx.doi.org/10.1016/j.jsv.2021.116196

Janiesch C. ; Zschech P. ; Heinrich K. Machine learning and deep learning . Electron. Mark. , 2021 , 31 ( 3 ), 685 - 695 . doi: 10.1007/s12525-021-00475-2 http://dx.doi.org/10.1007/s12525-021-00475-2

Dong S. ; Wang P. ; Abbas K. A survey on deep learning and its applications . Comput. Sci. Rev. , 2021 , 40 , 100379 . doi: 10.1016/j.cosrev.2021.100379 http://dx.doi.org/10.1016/j.cosrev.2021.100379

Das S. ; Tesfamariam S. State-of-the-art review of design of experiments for physics-informed deep learning . 2022 : arXiv: 2202. 06416 . https://arxiv.org/abs/2202.06416

Wang S. F. ; Sankaran S. ; Wang H. W. ; Perdikaris P. An expert's guide to training physics-informed neural networks . 2023 : arXiv: 2308. 08468 . https://arxiv.org/abs/2308.08468. doi: 10.1016/j.cma.2024.116813 http://dx.doi.org/10.1016/j.cma.2024.116813

Bilodeau C. ; Kazakov A. ; Mukhopadhyay S. ; Emerson J. ; Kalantar T. ; Muzny C. ; Jensen K. Machine learning for predicting the viscosity of binary liquid mixtures . Chem. Eng. J. , 2023 , 464 , 142454 . doi: 10.1016/j.cej.2023.142454 http://dx.doi.org/10.1016/j.cej.2023.142454

Nazar S. ; Yang J. ; Thomas B. S. ; Azim I. ; Ur Rehman S. K. Rheological properties of cementitious composites with and without nano-materials: a comprehensive review . J. Clean. Prod. , 2020 , 272 , 122701 . doi: 10.1016/j.jclepro.2020.122701 http://dx.doi.org/10.1016/j.jclepro.2020.122701

Khan W. A. ; Farooq S. ; Kadry S. ; Hanif M. ; Iftikhar F. J. ; Abbas S. Z. Variable characteristics of viscosity and thermal conductivity in peristalsis of magneto-Carreau nanoliquid with heat transfer irreversibilities . Comput. Meth. Programs Biomed. , 2020 , 190 , 105355 . doi: 10.1016/j.cmpb.2020.105355 http://dx.doi.org/10.1016/j.cmpb.2020.105355

Jin X. W. ; Cai S. Z. ; Li H. ; Karniadakis G. E. NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations . J. Comput. Phys. , 2021 , 426 , 109951 . doi: 10.1016/j.jcp.2020.109951 http://dx.doi.org/10.1016/j.jcp.2020.109951

Kimondo J. J. ; Said R. R. ; Wu J. ; Tian C. ; Wu Z. Mechanical rheological model on the assessment of elasticity and viscosity in tissue inflammation: a systematic review . PLoS One , 2024 , 19 ( 7 ), e 0307113 . doi: 10.1371/journal.pone.0307113 http://dx.doi.org/10.1371/journal.pone.0307113

Ling S. B. ; Wu Z. ; Mei J. Comparison and review of classical and machine learning-based constitutive models for polymers used in aeronautical thermoplastic composites . Rev. Adv. Mater. Sci. , 2023 , 62 , 20230107 . doi: 10.1515/rams-2023-0107 http://dx.doi.org/10.1515/rams-2023-0107

Abueidda D. W. ; Koric S. ; Abu Al-Rub R. ; Parrott C. M. ; James K. A. ; Sobh N. A. A deep learning energy method for hyperelasticity and viscoelasticity . Eur. J. Mech. A/solids , 2022 , 95 , 104639 . doi: 10.1016/j.euromechsol.2022.104639 http://dx.doi.org/10.1016/j.euromechsol.2022.104639

Chen X. X. ; Zhang P. ; Yu H. S. ; Yin Z. Y. ; Sheil B. Parsimonious universal function approximator for elastic and elastoplastic cavity expansion problems . J. Geotech. Geoenviron. Eng. , 2025 , 151 ( 9 ), 04025093 . doi: 10.1061/jggefk.gteng-13267 http://dx.doi.org/10.1061/jggefk.gteng-13267

Tian Y. J. ; Zhang Y. Q. A comprehensive survey on regularization strategies in machine learning . Inf. Fusion , 2022 , 80 , 146 - 166 . doi: 10.1016/j.inffus.2021.11.005 http://dx.doi.org/10.1016/j.inffus.2021.11.005

Wang M. H. ; Wang Q. ; Chanussot J. ; Hong D. F. L₀-l₁ hybrid total variation regularization and its applications on hyperspectral image mixed noise removal and compressed sensing . IEEE Trans. Geosci. Remote. Sens. , 2021 , 59 ( 9 ), 7695 - 7710 . doi: 10.1109/tgrs.2021.3055516 http://dx.doi.org/10.1109/tgrs.2021.3055516

Florez W. F. ; Popov V. ; Gaviria-Cardona J. P. ; Bustamante C. A. ; Martínez-Tejada H. V. ; Garcia-Tamayo E. A local collocation method with radial basis functions for an electrospinning problem . Eng. Anal. Bound. Elem. , 2022 , 134 , 398 - 411 . doi: 10.1016/j.enganabound.2021.10.013 http://dx.doi.org/10.1016/j.enganabound.2021.10.013

Rüttgers A. ; Griebel M. Multiscale simulation of polymeric fluids using the sparse grid combination technique . Appl. Math. Comput. , 2018 , 319 , 425 - 443 . doi: 10.1016/j.amc.2017.04.025 http://dx.doi.org/10.1016/j.amc.2017.04.025

Wang B. Y. ; Deng L. ; Qu Z. ; Li S. C. ; Zhang Z. ; Xie Y. Efficient processing of sparse tensor decomposition via unified abstraction and PE-interactive architecture . IEEE Trans. Comput. , 2022 , 71 ( 2 ), 266 - 281 . doi: 10.1109/tc.2020.3046617 http://dx.doi.org/10.1109/tc.2020.3046617

Messenger D. A. ; Bortz D. M. Weak SINDy for partial differential equations . J. Comput. Phys. , 2021 , 443 , 110525 . doi: 10.1016/j.jcp.2021.110525 http://dx.doi.org/10.1016/j.jcp.2021.110525

Bai J. S. ; Rabczuk T. ; Gupta A. ; Alzubaidi L. ; Gu Y. T. A physics-informed neural network technique based on a modified loss function for computational 2 D and 3 D solid mechanics . Comput. Mech., 2022, 71 ( 3 ), 543 - 562 . doi: 10.1007/s00466-022-02252-0 http://dx.doi.org/10.1007/s00466-022-02252-0

Thakur S. ; Mitra H. ; Ardekani A. M. Physics-informed neural network-based inverse framework for time-fractional differential equations for rheology . Biology , 2025 , 14 ( 7 ), 779 . doi: 10.3390/biology14070779 http://dx.doi.org/10.3390/biology14070779

Wang X. L. ; Jin Y. C. ; Schmitt S. ; Olhofer M. Recent advances in Bayesian optimization . ACM Comput. Surv. , 2023 , 55 ( 13 s), 1 - 36 . doi: 10.1145/3582078 http://dx.doi.org/10.1145/3582078

Khatamsaz D. ; Neuberger R. ; Roy A. M. ; Zadeh S. H. ; Otis R. ; Arróyave R. A physics informed Bayesian optimization approach for material design: application to NiTi shape memory alloys . NPJ Comput. Mater. , 2023 , 9 , 221 . doi: 10.1038/s41524-023-01173-7 http://dx.doi.org/10.1038/s41524-023-01173-7

Zhao X. G. ; Shirvan K. ; Salko R. K. ; Guo F. D. On the prediction of critical heat flux using a physics-informed machine learning-aided framework . Appl. Therm. Eng. , 2020 , 164 , 114540 . doi: 10.1016/j.applthermaleng.2019.114540 http://dx.doi.org/10.1016/j.applthermaleng.2019.114540

Kapoor T. ; Chandra A. ; Tartakovsky D. M. ; Wang H. R. ; Nunez A. ; Dollevoet R. Neural oscillators for generalization of physics-informed machine learning . Proc. AAAI Conf. Artif. Intell. , 2024 , 38 ( 12 ), 13059 - 13067 . doi: 10.1609/aaai.v38i12.29204 http://dx.doi.org/10.1609/aaai.v38i12.29204

Ahn K. H. ; Jamali S. Data-driven methods in rheology . Rheol. Acta , 2023 , 62 ( 10 ), 473 - 475 . doi: 10.1007/s00397-023-01416-w http://dx.doi.org/10.1007/s00397-023-01416-w

Shirangi M. G. ; Aragall R. ; Osgouei R. E. ; May R. ; Furlong E. ; Dahl T. G. ; Thompson C. A. Development of digital twins for drilling fluids: local velocities for hole cleaning and rheology monitoring . J. Energy Resour. Technol. , 2022 , 144 ( 12 ), 123003 . doi: 10.1115/1.4054463 http://dx.doi.org/10.1115/1.4054463

Butt J. ; Mohaghegh V. Combining digital twin and machine learning for the fused filament fabrication process . Metals , 2023 , 13 ( 1 ), 24 . doi: 10.3390/met13010024 http://dx.doi.org/10.3390/met13010024

Mahmoudabadbozchelou M. ; Kamani K. M. ; Rogers S. A. ; Jamali S. Digital rheometer twins: learning the hidden rheology of complex fluids through rheology-informed graph neural networks . Proc. Natl. Acad. Sci. U.S.A. , 2022 , 119 ( 20 ), e 2202234119 . doi: 10.1073/pnas.2202234119 http://dx.doi.org/10.1073/pnas.2202234119

Abadi M. , Barham P. , Chen J. , Chen Z. , Davis A. , Dean J. , Devin M. , Ghemawat S. , Irving G. , Isard M. , Kudlur M. , Levenberg J. , Monga R. , Moore S. , Murray D. G. , Steiner B. , Tucker P. , Vasudevan V. , Warden P. , Wicke M. , Yu Y. , Zheng X. TensorFlow: A System for Large-Scale Machine Learning . in : 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16 ). 2016 : 265 - 283 .

Paszke A. ; Gross S. ; Massa F. ; Lerer A. ; Bradbury J. ; Chanan G. ; Killeen T. ; Lin Z. M. ; Gimelshein N. ; Antiga L. ; Desmaison A. ; Köpf A. ; Yang E. ; DeVito Z. ; Raison M. ; Tejani A. ; Chilamkurthy S. ; Steiner B. ; Fang L. ; Bai J. J. ; Chintala S. PyTorch: an imperative style, high-performance deep learning library . 2019 , arXiv: 1912.01703 .

Ketkar N. Introduction to keras . Deep Learning with Python . Berkeley, CA: Apress, 2017 , 97 - 111 . doi: 10.1007/978-1-4842-2766-4_7 http://dx.doi.org/10.1007/978-1-4842-2766-4_7

Ma Y. ; Yu D. ; Wu T. ; Wang H. PaddlePaddle: an open-source deep learning platform from industrial practice . Front. Data Domputing , 2019 , 1 ( 1 ), 105 - 115 .

Lu L. ; Meng X. H. ; Mao Z. P. ; Karniadakis G. E. DeepXDE: a deep learning library for solving differential equations . SIAM Rev. , 2021 , 63 ( 1 ), 208 - 228 . doi: 10.1137/19m1274067 http://dx.doi.org/10.1137/19m1274067

Hennigh O. ; Narasimhan S. ; Nabian M. A. ; Subramaniam A. ; Tangsali K. ; Fang Z. W. ; Rietmann M. ; Byeon W. ; Choudhry S. NVIDIA SimNet TM : an AI-accelerated multi-physics simulation framework . Computational Science—ICCS 2021 . Cham : Springer International Publishing , 2021 , 447 - 461 . doi: 10.1007/978-3-030-77977-1_36 http://dx.doi.org/10.1007/978-3-030-77977-1_36

Koryagin A. ; Khudorozkov R. ; Tsimfer S. PyDEns: a Python framework for solving differential equations with neural networks . 2019 : arXiv: 1909. 11544 . https://arxiv.org/abs/1909.11544

Chen F. Y. ; Sondak D. ; Protopapas P. ; Mattheakis M. ; Liu S. H. ; Agarwal D. ; Di Giovanni M. NeuroDiffEq: a Python package for solving differential equations with neural networks . J. Open Source Softw. , 2020 , 5 ( 46 ), 1931 . doi: 10.21105/joss.01931 http://dx.doi.org/10.21105/joss.01931

Jospin L. V. ; Laga H. ; Boussaid F. ; Buntine W. ; Bennamoun M. Hands-on Bayesian neural networks: a tutorial for deep learning users . IEEE Comput. Intell. Mag. , 2022 , 17 ( 2 ), 29 - 48 . doi: 10.1109/mci.2022.3155327 http://dx.doi.org/10.1109/mci.2022.3155327

Yang L. ; Meng X. H. ; Karniadakis G. E. B-PINNs: bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data . J. Comput. Phys. , 2021 , 425 , 109913 . doi: 10.1016/j.jcp.2020.109913 http://dx.doi.org/10.1016/j.jcp.2020.109913

Linka K. ; Schäfer A. ; Meng X. H. ; Zou Z. R. ; Karniadakis G. E. ; Kuhl E. Bayesian Physics Informed Neural Networks for real-world nonlinear dynamical systems . Comput. Meth. Appl. Mech. Eng. , 2022 , 402 , 115346 . doi: 10.1016/j.cma.2022.115346 http://dx.doi.org/10.1016/j.cma.2022.115346

Sun L. N. ; Wang J. X. Physics-constrained Bayesian neural network for fluid flow reconstruction with sparse and noisy data . Theor. Appl. Mech. Lett. , 2020 , 10 ( 3 ), 161 - 169 . doi: 10.1016/j.taml.2020.01.031 http://dx.doi.org/10.1016/j.taml.2020.01.031

Meng X. H. ; Babaee H. ; Karniadakis G. E. Multi-fidelity Bayesian neural networks: algorithms and applications . J. Comput. Phys. , 2021 , 438 , 110361 . doi: 10.1016/j.jcp.2021.110361 http://dx.doi.org/10.1016/j.jcp.2021.110361

Ji B. ; Sharma Bhattarai S. ; Na I. H. ; Kim H. A Bayesian deep learning approach for rheological properties prediction of asphalt binders considering uncertainty of output . Constr. Build. Mater. , 2023 , 408 , 133671 . doi: 10.1016/j.conbuildmat.2023.133671 http://dx.doi.org/10.1016/j.conbuildmat.2023.133671

Zhao L. F. ; Li Z. ; Wang Z. C. ; Caswell B. ; Ouyang J. ; Karniadakis G. E. Active- and transfer-learning applied to microscale-macroscale coupling to simulate viscoelastic flows . J. Comput. Phys. , 2021 , 427 , 110069 . doi: 10.1016/j.jcp.2020.110069 http://dx.doi.org/10.1016/j.jcp.2020.110069

Hall E. J. ; Taverniers S. ; Katsoulakis M. A. ; Tartakovsky D. M. GINNs: graph-informed neural networks for multiscale physics . J. Comput. Phys. , 2021 , 433 , 110192 . doi: 10.1016/j.jcp.2021.110192 http://dx.doi.org/10.1016/j.jcp.2021.110192

Xing Z. Y. A non-Markov chain model of heterogeneous polymer dynamics for exploring relaxation and creep behavior . J. Phys. D: Appl. Phys. , 2025 , 58 ( 11 ), 115305 . doi: 10.1088/1361-6463/ada80c http://dx.doi.org/10.1088/1361-6463/ada80c

Gm H. ; Gourisaria M. K. ; Pandey M. ; Rautaray S. S. A comprehensive survey and analysis of generative models in machine learning . Comput. Sci. Rev. , 2020 , 38 , 100285 . doi: 10.1016/j.cosrev.2020.100285 http://dx.doi.org/10.1016/j.cosrev.2020.100285

Ghojogh B. ; Ghodsi A. ; Karray F. ; Crowley M. Factor analysis, probabilistic principal component analysis, variational inference, and variational autoencoder: tutorial and survey . 2021 , arXiv: 2101.00734 .

Gundersen K. ; Oleynik A. ; Blaser N. ; Alendal G. Semi-conditional variational auto-encoder for flow reconstruction and uncertainty quantification from limited observations . Phys. Fluids , 2021 , 33 , 017119 . doi: 10.1063/5.0025779 http://dx.doi.org/10.1063/5.0025779

Akkari N. ; Casenave F. ; Hachem E. ; Ryckelynck D. A Bayesian nonlinear reduced order modeling using variational AutoEncoders . Fluids , 2022 , 7 ( 10 ), 334 . doi: 10.3390/fluids7100334 http://dx.doi.org/10.3390/fluids7100334

Simpson T. ; Vlachas K. ; Garland A. ; Dervilis N. ; Chatzi E. VpROM: a novel variational autoencoder-boosted reduced order model for the treatment of parametric dependencies in nonlinear systems . Sci. Rep. , 2024 , 14 , 6091 . doi: 10.1038/s41598-024-56118-x http://dx.doi.org/10.1038/s41598-024-56118-x

Hora G. S. ; Gentine P. ; Momen M. ; Giometto M. G. Physics-informed data-driven reconstruction of turbulent wall-bounded flows from planar measurements . Phys. Fluids , 2024 , 36 ( 11 ), 115191 . doi: 10.1063/5.0239163 http://dx.doi.org/10.1063/5.0239163

Zhong, W. H.; Meidani, H. PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations . Comput. Meth. Appl. Mech. Eng. , 2023 , 403 , 115664 . doi: 10.1016/j.cma.2022.115664 http://dx.doi.org/10.1016/j.cma.2022.115664

Shin H. ; Choi M. Physics-informed variational inference for uncertainty quantification of stochastic differential equations . J. Comput. Phys. , 2023 , 487 , 112183 . doi: 10.1016/j.jcp.2023.112183 http://dx.doi.org/10.1016/j.jcp.2023.112183

Goodfellow I. ; Pouget-Abadie J. ; Mirza M. ; Xu B. ; Warde-Farley D. ; Ozair S. ; Courville A. ; Bengio Y. Generative adversarial networks . Commun. ACM , 2020 , 63 ( 11 ), 139 - 144 . doi: 10.1145/3422622 http://dx.doi.org/10.1145/3422622

Guo X. ; Chen Y. Q. Generative AI for synthetic data generation: Methods, challenges and the future . 2024 : arXiv: 2403. 04190 . https://arxiv.org/abs/2403.04190

Wang Z. Z. ; Jeong H. ; Gan Y. X. ; Pereira J. M. ; Gu Y. T. ; Sauret E. Pore-scale modeling of multiphase flow in porous media using a conditional generative adversarial network (cGAN) . Phys. Fluids , 2022 , 34 ( 12 ), 123325 . doi: 10.1063/5.0133054 http://dx.doi.org/10.1063/5.0133054

Xie P. X. ; Li R. ; Chen Y. R. ; Song B. Y. ; Chen W. L. ; Zhou D. ; Cao Y. A physics-informed deep learning model to reconstruct turbulent wake from random sparse data . Phys. Fluids , 2024 , 36 ( 6 ), 065145 . doi: 10.1063/5.0212298 http://dx.doi.org/10.1063/5.0212298

Cheng M. ; Fang F. ; Pain C. C. ; Navon I. M. An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling . Comput. Meth. Appl. Mech. Eng. , 2020 , 372 , 113375 . doi: 10.1016/j.cma.2020.113375 http://dx.doi.org/10.1016/j.cma.2020.113375

Niu Z. J. ; Yu K. ; Wu X. F. LSTM-based VAE-GAN for time-series anomaly detection . Sensors , 2020 , 20 ( 13 ), 3738 . doi: 10.3390/s20133738 http://dx.doi.org/10.3390/s20133738

Jiang Y. ; Liang Y. F. ; Yuan X. F. Super-resolution reconstruction of turbulence for Newtonian and viscoelastic fluids with a physical constraint . Phys. Fluids , 2024 , 36 ( 10 ), 103114 . doi: 10.1063/5.0232224 http://dx.doi.org/10.1063/5.0232224

Wang Y. J. ; Shan L. Y. ; Yan J. M. ; Pang Y. Z. A data augmentation method for establishing a relationship model between composition and viscoelastic properties of asphalt binder . IEEE Trans. Intell. Transp. Syst. , 2025 , 26 ( 8 ), 12446 - 12460 . doi: 10.1109/tits.2025.3558911 http://dx.doi.org/10.1109/tits.2025.3558911

Lu L. ; Jin P. Z. ; Pang G. F. ; Zhang Z. Q. ; Karniadakis G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators . Nat. Mach. Intell. , 2021 , 3 ( 3 ), 218 - 229 . doi: 10.1038/s42256-021-00302-5 http://dx.doi.org/10.1038/s42256-021-00302-5

Wang S. F. ; Wang H. W. ; Perdikaris P. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets . Sci. Adv. , 2021 , 7 ( 40 ), eabi 8605 . doi: 10.1126/sciadv.abi8605 http://dx.doi.org/10.1126/sciadv.abi8605

Kovachki N. ; Li Z. Y. ; Liu B. ; Azizzadenesheli K. ; Bhattacharya K. ; Stuart A. ; Anandkumar A. Neural operator: learning maps between function spaces with applications to PDEs . J. Mach. Learn. Res. , 2023 , 24 ( 1 ), 4061 - 4157 .

Li Z. Y. ; Kovachki N. ; Azizzadenesheli K. ; Liu B. ; Bhattacharya K. ; Stuart A. ; Anandkumar A. Fourier neural operator for parametric partial differential equations . 2020 : arXiv: 2010. 08895 . https://arxiv.org/abs/2010.08895

Li Z. Y. ; Zheng H. K. ; Kovachki N. ; Jin D. ; Chen H. X. ; Liu B. ; Azizzadenesheli K. ; Anandkumar A. Physics-informed neural operator for learning partial differential equations . ACM / IMS J. Data Sci. , 2024 , 1 ( 3 ), 1 - 27 . doi: 10.1145/3648506 http://dx.doi.org/10.1145/3648506

Cao Q. Y. ; Goswami S. ; Karniadakis G. E. Laplace neural operator for solving differential equations . Nat. Mach. Intell. , 2024 , 6 ( 6 ), 631 - 640 . doi: 10.1038/s42256-024-00844-4 http://dx.doi.org/10.1038/s42256-024-00844-4

Lowery M. ; Turnage J. ; Morrow Z. ; Jakeman J. D. ; Narayan A. ; Zhe S. D. ; Shankar V. Kernel neural operators (KNOs) for scalable, memory-efficient, geometrically-flexible operator learning . 2024 : arXiv: 2407. 00809 . https://arxiv.org/abs/2407.00809

Jeyaraj D. ; Eivazi H. ; Tröger J. A. ; Wittek S. ; Hartmann S. ; Rausch A. A neural operator based hybrid microscale model for multiscale simulation of rate-dependent materials . 2025 : arXiv: 2506. 16918 . https://arxiv.org/abs/2506.16918

Mangal D. ; Saadat M. ; Jamali S. Learning a family of rheological constitutive models using neural operators . J. Rheol. , 2025 , 69 ( 2 ), 55 - 67 . doi: 10.1122/8.0000908 http://dx.doi.org/10.1122/8.0000908

Zhou J. ; Cui G. Q. ; Hu S. D. ; Zhang Z. Y. ; Yang C. ; Liu Z. Y. ; Wang L. F. ; Li C. C. ; Sun M. S. Graph neural networks: a review of methods and applications . AI Open , 2020 , 1 , 57 - 81 . doi: 10.1016/j.aiopen.2021.01.001 http://dx.doi.org/10.1016/j.aiopen.2021.01.001

Liu S. C. ; Li T. ; Ding H. Y. ; Tang B. Z. ; Wang X. L. ; Chen Q. C. ; Yan J. ; Zhou Y. A hybrid method of recurrent neural network and graph neural network for next-period prescription prediction . Int. J. Mach. Learn. Cybern. , 2020 , 11 ( 12 ), 2849 - 2856 . doi: 10.1007/s13042-020-01155-x http://dx.doi.org/10.1007/s13042-020-01155-x

Atkinson O. ; Bhardwaj A. ; Englert C. ; Ngairangbam V. S. ; Spannowsky M. Anomaly detection with convolutional graph neural networks . J. High Energy Phys. , 2021 , 2021 ( 8 ), 80 . doi: 10.1007/jhep08(2021)080 http://dx.doi.org/10.1007/jhep08(2021)080

Du X. S. ; Yu J. ; Chu Z. ; Jin L. N. ; Chen J. Y. Graph autoencoder-based unsupervised outlier detection . Inf. Sci. , 2022 , 608 , 532 - 550 . doi: 10.1016/j.ins.2022.06.039 http://dx.doi.org/10.1016/j.ins.2022.06.039

Jin G. Y. ; Liang Y. X. ; Fang Y. C. ; Shao Z. Z. ; Huang J. C. ; Zhang J. B. ; Zheng Y. Spatio-temporal graph neural networks for predictive learning in urban computing: a survey . IEEE Trans. Knowl. Data Eng. , 2024 , 36 ( 10 ), 5388 - 5408 . doi: 10.1109/tkde.2023.3333824 http://dx.doi.org/10.1109/tkde.2023.3333824

Li Z. J. ; Farimani A. B. Graph neural network-accelerated Lagrangian fluid simulation . Comput. Graph. , 2022 , 103 , 201 - 211 . doi: 10.1016/j.cag.2022.02.004 http://dx.doi.org/10.1016/j.cag.2022.02.004

Jiang S. L. ; Dieng A. B. ; Webb M. A. Property-guided generation of complex polymer topologies using variational autoencoders . NPJ Comput. Mater. , 2024 , 10 , 139 . doi: 10.1038/s41524-024-01328-0 http://dx.doi.org/10.1038/s41524-024-01328-0

Shukla K. ; Xu M. J. ; Trask N. ; Karniadakis G. E. Scalable algorithms for physics-informed neural and graph networks . Data Centric Eng. , 2022 , 3 , e24 . doi: 10.1017/dce.2022.24 http://dx.doi.org/10.1017/dce.2022.24

Gao H. ; Zahr M. J. ; Wang J. X. Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems . Comput. Meth. Appl. Mech. Eng. , 2022 , 390 , 114502 . doi: 10.1016/j.cma.2021.114502 http://dx.doi.org/10.1016/j.cma.2021.114502

Li Z. Y. ; Kovachki N. ; Azizzadenesheli K. ; Liu B. ; Bhattacharya K. ; Stuart A. ; Anandkumar A. Multipole graph neural operator for parametric partial differential equations . 2020 : arXiv: 2006. 09535 . https://arxiv.org/abs/2006.09535

Storm J. ; Rocha I. B. C. M. ; van der Meer F. P. A microstructure-based graph neural network for accelerating multiscale simulations . Comput. Meth. Appl. Mech. Eng. , 2024 , 427 , 117001 . doi: 10.1016/j.cma.2024.117001 http://dx.doi.org/10.1016/j.cma.2024.117001

Du W. ; Ding S. F. A survey on multi-agent deep reinforcement learning: from the perspective of challenges and applications . Artif. Intell. Rev. , 2021 , 54 ( 5 ), 3215 - 3238 . doi: 10.1007/s10462-020-09938-y http://dx.doi.org/10.1007/s10462-020-09938-y

Garnier P. ; Viquerat J. ; Rabault J. ; Larcher A. ; Kuhnle A. ; Hachem E. A review on deep reinforcement learning for fluid mechanics . Comput. Fluids , 2021 , 225 , 104973 . doi: 10.1016/j.compfluid.2021.104973 http://dx.doi.org/10.1016/j.compfluid.2021.104973

Banerjee C. ; Nguyen K. ; Fookes C. ; Raissi M. A survey on physics informed reinforcement learning: review and open problems . Expert Syst. Appl. , 2025 , 287 , 128166 . doi: 10.1016/j.eswa.2025.128166 http://dx.doi.org/10.1016/j.eswa.2025.128166

Liu X. Y. ; Wang J. X. Physics-informed Dyna-style model-based deep reinforcement learning for dynamic control . Proc. Math. Phys. Eng. Sci. , 2021 , 477 ( 2255 ), 1 - 24 . doi: 10.1098/rspa.2021.0618 http://dx.doi.org/10.1098/rspa.2021.0618

Hu W. L. ; Jiang Z. Z. ; Xu M. Y. ; Hu H. Y. Efficient deep reinforcement learning strategies for active flow control based on physics-informed neural networks . Phys. Fluids , 2024 , 36 ( 7 ), 074112 . doi: 10.1063/5.0213256 http://dx.doi.org/10.1063/5.0213256

Faria R. R. ; Capron B. D. O. ; Secchi A. R. ; De Souza M. B. A data-driven tracking control framework using physics-informed neural networks and deep reinforcement learning for dynamical systems . Eng. Appl. Artif. Intell. , 2024 , 127 , 107256 . doi: 10.1016/j.engappai.2023.107256 http://dx.doi.org/10.1016/j.engappai.2023.107256

San O. ; Pawar S. ; Rasheed A. Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems . Sci. Rep. , 2022 , 12 , 17947 . doi: 10.1038/s41598-022-22598-y http://dx.doi.org/10.1038/s41598-022-22598-y

Gyimah N. ; Scheler O. ; Rang T. ; Pardy T. Deep reinforcement learning-based digital twin for droplet microfluidics control . Phys. Fluids , 2023 , 35 ( 8 ), 082020 . doi: 10.1063/5.0159981 http://dx.doi.org/10.1063/5.0159981

Fuks O. ; Tchelepi H. A. Limitations of physics informed machine learning for nonlinear two-phase transport in porous media . J. Mach. Learn. Model. Comput. , 2020 , 1 ( 1 ), 19 - 37 . doi: 10.1615/jmachlearnmodelcomput.2020033905 http://dx.doi.org/10.1615/jmachlearnmodelcomput.2020033905

Miyamoto S. Short review on machine learning-based multi-scale simulation in rheology . J. Soc. Rheol. , Jpn, 2024 , 52 ( 1 ), 15 - 19 . doi: 10.1678/rheology.52.15 http://dx.doi.org/10.1678/rheology.52.15

浏览量

下载量

CSCD

文章被引用时，请邮件提醒。

提交

工具集

关联资源

暂无数据