Year: 2019
Author: Chao Ma, Jianchun Wang, Weinan E
Communications in Computational Physics, Vol. 25 (2019), Iss. 4 : pp. 947–962
Abstract
The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced model. In this work, we explore a natural analogy between recurrent neural networks and the Mori-Zwanzig formalism to establish a systematic approach for developing reduced models with memory. Two training models — a direct training model and a dynamically coupled training model — are proposed and compared. We apply these methods to the Kuramoto-Sivashinsky equation and the Navier-Stokes equation. Numerical experiments show that the proposed method can produce reduced model with good performance on both short-term prediction and long-term statistical properties.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2018-0269
Communications in Computational Physics, Vol. 25 (2019), Iss. 4 : pp. 947–962
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Model reduction Mori-Zwanzig recurrent neural networks.
Author Details
-
Data-driven stochastic closure modeling via conditional diffusion model and neural operator
Dong, Xinghao | Chen, Chuanqi | Wu, Jin-LongJournal of Computational Physics, Vol. 534 (2025), Iss. P.114005
https://doi.org/10.1016/j.jcp.2025.114005 [Citations: 0] -
LEARNING EPIDEMIOLOGICAL DYNAMICS VIA THE FINITE EXPRESSION METHOD
Du, Jianda | Liang, Senwei | Wang, ChunmeiJournal of Machine Learning for Modeling and Computing, Vol. 6 (2025), Iss. 1 P.83
https://doi.org/10.1615/JMachLearnModelComput.2025057903 [Citations: 0] -
Temporally sparse data assimilation for the small-scale reconstruction of turbulence
Wang, Yunpeng | Yuan, Zelong | Xie, Chenyue | Wang, JianchunPhysics of Fluids, Vol. 34 (2022), Iss. 6
https://doi.org/10.1063/5.0091391 [Citations: 11] -
Artificial neural network approach for turbulence models: A local framework
Xie, Chenyue | Xiong, Xiangming | Wang, JianchunPhysical Review Fluids, Vol. 6 (2021), Iss. 8
https://doi.org/10.1103/PhysRevFluids.6.084612 [Citations: 18] -
The effect of filter anisotropy on the large eddy simulation of turbulence
Chang, Ning | Yuan, Zelong | Wang, Yunpeng | Wang, JianchunPhysics of Fluids, Vol. 35 (2023), Iss. 3
https://doi.org/10.1063/5.0142643 [Citations: 2] -
Deconvolutional artificial-neural-network framework for subfilter-scale models of compressible turbulence
Yuan, Zelong | Wang, Yunpeng | Xie, Chenyue | Wang, JianchunActa Mechanica Sinica, Vol. 37 (2021), Iss. 12 P.1773
https://doi.org/10.1007/s10409-021-01150-7 [Citations: 17] -
Artificial neural network-based nonlinear algebraic models for large eddy simulation of turbulence
Xie, Chenyue | Yuan, Zelong | Wang, JianchunPhysics of Fluids, Vol. 32 (2020), Iss. 11
https://doi.org/10.1063/5.0025138 [Citations: 72] -
Memory-based parameterization with differentiable solver: Application to Lorenz ’96
Bhouri, Mohamed Aziz | Gentine, PierreChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 33 (2023), Iss. 7
https://doi.org/10.1063/5.0131929 [Citations: 3] -
Learning Proper Orthogonal Decomposition of Complex Dynamics Using Heavy-ball Neural ODEs
Baker, Justin | Cherkaev, Elena | Narayan, Akil | Wang, BaoJournal of Scientific Computing, Vol. 95 (2023), Iss. 2
https://doi.org/10.1007/s10915-023-02176-8 [Citations: 2] -
Thermodynamics of Learning Physical Phenomena
Cueto, Elias | Chinesta, FranciscoArchives of Computational Methods in Engineering, Vol. 30 (2023), Iss. 8 P.4653
https://doi.org/10.1007/s11831-023-09954-5 [Citations: 14] -
On the structure of time-delay embedding in linear models of non-linear dynamical systems
Pan, Shaowu | Duraisamy, KarthikChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 30 (2020), Iss. 7
https://doi.org/10.1063/5.0010886 [Citations: 54] -
Modeling subgrid-scale forces by spatial artificial neural networks in large eddy simulation of turbulence
Xie, Chenyue | Wang, Jianchun | E, WeinanPhysical Review Fluids, Vol. 5 (2020), Iss. 5
https://doi.org/10.1103/PhysRevFluids.5.054606 [Citations: 95] -
Dynamically learning the parameters of a chaotic system using partial observations
Carlson, Elizabeth | Hudson, Joshua | Larios, Adam | Martinez, Vincent R. | Ng, Eunice | Whitehead, Jared P.Discrete and Continuous Dynamical Systems, Vol. 42 (2022), Iss. 8 P.3809
https://doi.org/10.3934/dcds.2022033 [Citations: 17] -
Overview Frequency Principle/Spectral Bias in Deep Learning
Xu, Zhi-Qin John | Zhang, Yaoyu | Luo, TaoCommunications on Applied Mathematics and Computation, Vol. 7 (2025), Iss. 3 P.827
https://doi.org/10.1007/s42967-024-00398-7 [Citations: 8] -
Machine Learning in Modeling and Simulation
Reduced Order Modeling
Dar, Zulkeefal | Baiges, Joan | Codina, Ramon2023
https://doi.org/10.1007/978-3-031-36644-4_8 [Citations: 1] -
Toward Model Reduction for Power System Transients With Physics-Informed PDE
Pagnier, Laurent | Fritzsch, Julian | Jacquod, Philippe | Chertkov, MichaelIEEE Access, Vol. 10 (2022), Iss. P.65118
https://doi.org/10.1109/ACCESS.2022.3183336 [Citations: 7] -
OnsagerNet: Learning stable and interpretable dynamics using a generalized Onsager principle
Yu, Haijun | Tian, Xinyuan | E, Weinan | Li, QianxiaoPhysical Review Fluids, Vol. 6 (2021), Iss. 11
https://doi.org/10.1103/PhysRevFluids.6.114402 [Citations: 29] -
Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism
Wang, Qian | Ripamonti, Nicolò | Hesthaven, Jan S.Journal of Computational Physics, Vol. 410 (2020), Iss. P.109402
https://doi.org/10.1016/j.jcp.2020.109402 [Citations: 88] -
A framework for machine learning of model error in dynamical systems
Levine, Matthew | Stuart, AndrewCommunications of the American Mathematical Society, Vol. 2 (2022), Iss. 7 P.283
https://doi.org/10.1090/cams/10 [Citations: 33] -
Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto–Sivashinsky test case
Mojgani, Rambod | Chattopadhyay, Ashesh | Hassanzadeh, PedramChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32 (2022), Iss. 6
https://doi.org/10.1063/5.0091282 [Citations: 11] -
Artificial neural network-based spatial gradient models for large-eddy simulation of turbulence
Wang, Yunpeng | Yuan, Zelong | Xie, Chenyue | Wang, JianchunAIP Advances, Vol. 11 (2021), Iss. 5
https://doi.org/10.1063/5.0053590 [Citations: 36] -
Uniformly accurate machine learning-based hydrodynamic models for kinetic equations
Han, Jiequn | Ma, Chao | Ma, Zheng | E, WeinanProceedings of the National Academy of Sciences, Vol. 116 (2019), Iss. 44 P.21983
https://doi.org/10.1073/pnas.1909854116 [Citations: 68] -
Error bounds of the invariant statistics in machine learning of ergodic Itô diffusions
Zhang, He | Harlim, John | Li, XiantaoPhysica D: Nonlinear Phenomena, Vol. 427 (2021), Iss. P.133022
https://doi.org/10.1016/j.physd.2021.133022 [Citations: 5] -
Dynamic iterative approximate deconvolution models for large-eddy simulation of turbulence
Yuan, Zelong | Wang, Yunpeng | Xie, Chenyue | Wang, JianchunPhysics of Fluids, Vol. 33 (2021), Iss. 8
https://doi.org/10.1063/5.0059643 [Citations: 24] -
Data-driven construction of stochastic reduced dynamics encoded with non-Markovian features
She, Zhiyuan | Ge, Pei | Lei, HuanThe Journal of Chemical Physics, Vol. 158 (2023), Iss. 3
https://doi.org/10.1063/5.0130033 [Citations: 12] -
Spatial artificial neural network model for subgrid-scale stress and heat flux of compressible turbulence
Xie, Chenyue | Wang, Jianchun | Li, Hui | Wan, Minping | Chen, ShiyiTheoretical and Applied Mechanics Letters, Vol. 10 (2020), Iss. 1 P.27
https://doi.org/10.1016/j.taml.2020.01.006 [Citations: 25] -
Optimized CNN for Fault Recognition in Power Systems with High Penetrations of Inverter-Based Resources
Tant, Connor | Gokaraju, Ramakrishna2025 IEEE/IAS 61st Industrial and Commercial Power Systems Technical Conference (I&CPS), (2025), P.1
https://doi.org/10.1109/ICPS64254.2025.11030383 [Citations: 0] -
The effect of sub-filter scale dynamics in large eddy simulation of turbulence
Chang, Ning | Yuan, Zelong | Wang, JianchunPhysics of Fluids, Vol. 34 (2022), Iss. 9
https://doi.org/10.1063/5.0098925 [Citations: 10] -
Kernel-based prediction of non-Markovian time series
Gilani, Faheem | Giannakis, Dimitrios | Harlim, JohnPhysica D: Nonlinear Phenomena, Vol. 418 (2021), Iss. P.132829
https://doi.org/10.1016/j.physd.2020.132829 [Citations: 20] -
Deconvolutional artificial neural network models for large eddy simulation of turbulence
Yuan, Zelong | Xie, Chenyue | Wang, JianchunPhysics of Fluids, Vol. 32 (2020), Iss. 11
https://doi.org/10.1063/5.0027146 [Citations: 71] -
Extracting Interpretable Physical Parameters from Spatiotemporal Systems Using Unsupervised Learning
Lu, Peter Y. | Kim, Samuel | Soljačić, MarinPhysical Review X, Vol. 10 (2020), Iss. 3
https://doi.org/10.1103/PhysRevX.10.031056 [Citations: 26] -
Artificial neural network mixed model for large eddy simulation of compressible isotropic turbulence
Xie, Chenyue | Wang, Jianchun | Li, Hui | Wan, Minping | Chen, ShiyiPhysics of Fluids, Vol. 31 (2019), Iss. 8
https://doi.org/10.1063/1.5110788 [Citations: 79] -
Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence
Xie, Chenyue | Wang, Jianchun | Li, Ke | Ma, ChaoPhysical Review E, Vol. 99 (2019), Iss. 5
https://doi.org/10.1103/PhysRevE.99.053113 [Citations: 61] -
Data-driven discovery of quasiperiodically driven dynamics
Das, Suddhasattwa | Mustavee, Shakib | Agarwal, ShauryaNonlinear Dynamics, Vol. 113 (2025), Iss. 5 P.4097
https://doi.org/10.1007/s11071-024-09970-5 [Citations: 1] -
Shock trace prediction by reduced models for a viscous stochastic Burgers equation
Chen, Nan | Liu, Honghu | Lu, FeiChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32 (2022), Iss. 4
https://doi.org/10.1063/5.0084955 [Citations: 4] -
Spatially multi-scale artificial neural network model for large eddy simulation of compressible isotropic turbulence
Xie, Chenyue | Wang, Jianchun | Li, Hui | Wan, Minping | Chen, ShiyiAIP Advances, Vol. 10 (2020), Iss. 1
https://doi.org/10.1063/1.5138681 [Citations: 30] -
Simultaneous neural network approximation for smooth functions
Hon, Sean | Yang, HaizhaoNeural Networks, Vol. 154 (2022), Iss. P.152
https://doi.org/10.1016/j.neunet.2022.06.040 [Citations: 9] -
Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond
Lucarini, Valerio | Chekroun, Mickaël D.Nature Reviews Physics, Vol. 5 (2023), Iss. 12 P.744
https://doi.org/10.1038/s42254-023-00650-8 [Citations: 15] -
Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential
Grubišić, Luka | Hajba, Marko | Lacmanović, DomagojEntropy, Vol. 23 (2021), Iss. 1 P.95
https://doi.org/10.3390/e23010095 [Citations: 9] -
A Dynamical Systems Approach to Machine Learning
Pourmohammad Azizi, S. | Neisy, Abdolsadeh | Ahmad Waloo, SajadInternational Journal of Computational Methods, Vol. 20 (2023), Iss. 09
https://doi.org/10.1142/S021987622350007X [Citations: 3]