[Application de modèles d’ordre réduit pour des représentations temporelles multi-échelles dans la prédiction des systèmes dynamiques]
Modeling and predicting the dynamics of complex multiscale systems remains a significant challenge due to their inherent nonlinearities and sensitivity to initial conditions, as well as limitations of traditional machine learning methods that fail to capture high frequency behaviors. To overcome these difficulties, we propose three approaches for multiscale learning. The first leverages the Partition of Unity (PU) method, integrated with neural networks, to decompose the dynamics into local components and directly predict both macro- and micro-scale behaviors. The second applies the Singular Value Decomposition (SVD) to extract dominant modes that explicitly separate macro- and micro-scale dynamics. Since full access to the data matrix is rarely available in practice, we further employ a sparse high-order SVD to reconstruct multiscale dynamics from limited measurements. Together, these approaches ensure that both coarse and fine dynamics are accurately captured, making the framework effective for real-world applications involving complex, multi-scale phenomena and adaptable to higher-dimensional systems with incomplete observations, by providing an approximation and interpretation in all time scales present in the phenomena under study.
La modélisation et la prédiction de la dynamique des systèmes multi-échelles complexes restent un défi majeur en raison de leurs non-linéarités intrinsèques et de leur sensibilité aux conditions initiales, ainsi que des limites des méthodes d’apprentissage automatique traditionnelles qui ne parviennent pas à capturer les comportements à haute fréquence. Pour surmonter ces difficultés, nous proposons trois approches pour l’apprentissage multi-échelle. La première exploite la méthode de la Partition de l’Unité (PU), intégrée à des réseaux de neurones, afin de décomposer la dynamique en composantes locales et de prédire directement les comportements aux échelles macro et micro. La seconde applique la Décomposition en Valeurs Singulières (SVD) afin d’extraire des modes dominants qui séparent explicitement les dynamiques macro- et micro-échelles. Étant donné que l’accès complet à la matrice de données est rarement disponible en pratique, nous utilisons en outre une SVD d’ordre élevé parcimonieuse pour reconstruire la dynamique multi-échelle à partir d’un nombre limité de mesures. Ensemble, ces approches permettent de capturer avec précision à la fois les dynamiques grossières et fines, rendant le cadre proposé efficace pour des applications réelles impliquant des phénomènes complexes multi-échelles et adaptable à des systèmes de dimension plus élevée avec des observations incomplètes, en fournissant une approximation et une interprétation à toutes les échelles temporelles présentes dans le phénomène étudié.
Révisé le :
Accepté le :
Publié le :
Mots-clés : Dynamique multi-échelle, partition de l’unité, réseaux de neurones, décomposition en valeurs singulières, SVD d’ordre élevé parcimonieuse, réduction de modèle, modélisation basée sur les données
Elias Al Ghazal 1 ; Jad Mounayer 1 ; Beatriz Moya 1 ; Sebastian Rodriguez 1 ; Chady Ghnatios 2 ; Francisco Chinesta 1 , 3
CC-BY 4.0
@article{CRMECA_2026__354_G1_203_0,
author = {Elias Al Ghazal and Jad Mounayer and Beatriz Moya and Sebastian Rodriguez and Chady Ghnatios and Francisco Chinesta},
title = {Application of reduced-order models for temporal multiscale representations in the prediction of dynamical systems},
journal = {Comptes Rendus. M\'ecanique},
pages = {203--226},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {354},
doi = {10.5802/crmeca.355},
language = {en},
}
TY - JOUR AU - Elias Al Ghazal AU - Jad Mounayer AU - Beatriz Moya AU - Sebastian Rodriguez AU - Chady Ghnatios AU - Francisco Chinesta TI - Application of reduced-order models for temporal multiscale representations in the prediction of dynamical systems JO - Comptes Rendus. Mécanique PY - 2026 SP - 203 EP - 226 VL - 354 PB - Académie des sciences, Paris DO - 10.5802/crmeca.355 LA - en ID - CRMECA_2026__354_G1_203_0 ER -
%0 Journal Article %A Elias Al Ghazal %A Jad Mounayer %A Beatriz Moya %A Sebastian Rodriguez %A Chady Ghnatios %A Francisco Chinesta %T Application of reduced-order models for temporal multiscale representations in the prediction of dynamical systems %J Comptes Rendus. Mécanique %D 2026 %P 203-226 %V 354 %I Académie des sciences, Paris %R 10.5802/crmeca.355 %G en %F CRMECA_2026__354_G1_203_0
Elias Al Ghazal; Jad Mounayer; Beatriz Moya; Sebastian Rodriguez; Chady Ghnatios; Francisco Chinesta. Application of reduced-order models for temporal multiscale representations in the prediction of dynamical systems. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 203-226. doi: 10.5802/crmeca.355
[1] Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems (2017) | arXiv
[2] Multiscale Proper Generalized Decomposition Based on the Partition of Unity, Int. J. Numer. Methods Eng., Volume 120 (2019) no. 6, pp. 727-747 | DOI | Zbl | MR
[3] Learning Slow and Fast System Dynamics via Automatic Separation of Time Scales, Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’23) (Ambuj Singh; Yizhou Sun, eds.), ACM Press, 2023, pp. 4376-4386 | DOI
[4] Multiple scales analysis of slow-fast quasilinear systems, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci., Volume 475 (2019) no. 2223, 20180630, 17 pages | DOI | Zbl | MR
[5] The futures of climate modeling, npj Clim. Atmos. Sci., Volume 8 (2025), 99, 6 pages | DOI
[6] Sensitive Dependence on Initial Conditions of Strongly Nonlinear Periodic Orbits of the Forced Pendulum, Nonlinear Dyn., Volume 16 (1998) no. 3, pp. 223-237 | DOI | Zbl | MR
[7] Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., Volume 378 (2019), pp. 686-707 | DOI | Zbl | MR
[8] Automated discovery of fundamental variables hidden in experimental data, Nat. Comp. Sci, Volume 2 (2022) no. 7, pp. 433-442 | DOI
[9] Latent variable discovery in classification models, Artif. Intell. Med., Volume 30 (2004) no. 3, pp. 283-299 | DOI
[10] Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations, Comput. Methods Appl. Mech. Eng., Volume 432(A) (2024), 117442 | DOI | Zbl | MR
[11] LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems (2024) | arXiv
[12] Physics-informed machine learning, Nat. Rev. Phys., Volume 3 (2021) no. 6, pp. 422-440 | DOI
[13] A comparison of single and double generator formalisms for thermodynamics-informed neural networks, Comput. Mech., Volume 75 (2025) no. 6, pp. 1769-1785 | DOI | Zbl
[14] On the feasibility of foundational models for the simulation of physical phenomena, Int. J. Numer. Methods Eng., Volume 126 (2025) no. 6, e70027, 14 pages | Zbl
[15] Structure-preserving neural networks, J. Comput. Phys., Volume 426 (2021), 109950 | Zbl | MR
[16] Learning dynamical systems from data: An introduction to physics-guided deep learning, Proc. Natl. Acad. Sci. USA, Volume 121 (2024) no. 27, e2311808121, 10 pages | DOI | MR
[17] KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis, Proceedings of the 42nd International Conference on Machine Learning (Aarti Singh; Maryam Fazel; Daniel Hsu; Simon Lacoste-Julien; Felix Berkenkamp; Tegan Maharaj; Kiri Wagstaff; Jerry Zhu, eds.) (Proceedings of Machine Learning Research), Volume 267, PMLR, 2025, pp. 2484-2519
[18] Deep Koopman operator framework for causal discovery in nonlinear dynamical systems (2025) | arXiv
[19] Koopman operators for estimation and control of dynamical systems, Ann. Rev. Control Rob. Auton. Syst., Volume 4 (2021) no. 1, pp. 59-87 | DOI
[20] Learning Transformed Dynamics for Efficient Control Purposes, Mathematics, Volume 12 (2024) no. 14, 2251, 22 pages | DOI
[21] Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA, Volume 113 (2016) no. 15, pp. 3932-3937 | DOI | MR
[22] When are dynamical systems learned from time series data statistically accurate?, Adv. Neural Inf. Process. Syst., Volume 37 (2024), pp. 43975-44008
[23] D-FNO: A decomposed Fourier neural operator for large-scale parametric partial differential equations, Comput. Methods Appl. Mech. Eng., Volume 436 (2025), 117732 | MR
[24] Neural operator learning for long-time integration in dynamical systems with recurrent neural networks, 2024 International Joint Conference on Neural Networks (IJCNN) (Akira Hirose; Hisao Ishibuchi, eds.), IEEE Press, 2024, pp. 1-8
[25] Transfer learning-based physics-informed DeepONets for the adaptive evolution of digital twin models for dynamic systems, Nonlinear Dyn., Volume 113 (2025), pp. 19075-19102 | DOI
[26] Operator learning for predicting multiscale bubble growth dynamics, J. Chem. Phys., Volume 154 (2021) no. 10, 104118
[27] A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials, Comput. Methods Appl. Mech. Eng., Volume 391 (2022), 114587 | MR
[28] On the spectral bias of neural networks, International conference on machine learning (Kamalika Chaudhuri; Ruslan Salakhutdinov, eds.) (Proceedings of Machine Learning Research), Volume 97, PMLR, 2019, pp. 5301-5310
[29] Frequency principle: Fourier analysis sheds light on deep neural networks (2019) | arXiv
[30] U-FNO — An enhanced Fourier neural operator-based deep-learning model for multiphase flow, Adv. Water Resources, Volume 163 (2022), 104180
[31] A reduced simulation applied to viscoelastic fatigue of polymers using a time multi-scale approach based on partition of Unity method, Adv. Model. Simul. Eng. Sci., Volume 12 (2025) no. 1, 11, 16 pages | DOI
[32] A separated representation involving multiple time scales within the Proper Generalized Decomposition framework, Adv. Model. Simul. Eng. Sci., Volume 8 (2021) no. 1, 26, 22 pages | DOI
[33] A time multiscale based data-driven approach in cyclic elasto-plasticity, Comput. Struct., Volume 295 (2024), 107277, 14 pages | DOI
[34] Learning interpretable hierarchical dynamical systems models from time series data (2024) | arXiv
[35] Multi-scale deep neural network (MscaleDNN) for solving Poisson–Boltzmann equation in complex domains (2020) | arXiv
[36] Multi-scale simulation of complex systems: a perspective of integrating knowledge and data, ACM Comput. Surv., Volume 56 (2024) no. 12, pp. 1-38 | DOI
[37] Multiscale DeepONet for nonlinear operators in oscillatory function spaces for building seismic wave responses (2021) | arXiv
[38] A multifidelity deep operator network approach to closure for multiscale systems, Comput. Methods Appl. Mech. Eng., Volume 414 (2023), 116161 | MR
[39] Capturing the diffusive behavior of the multiscale linear transport equations by Asymptotic-Preserving Convolutional DeepONets, Comput. Methods Appl. Mech. Eng., Volume 418 (2024), 116531 | MR
[40] Structure-Preserving Model Reduction for Dissipative Mechanical Systems, Calm, Smooth and Smart (Peter Eberhard, ed.) (Lecture Notes in Computational Science and Engineering), Volume 102, Springer, 2023, pp. 209-230 | DOI
[41] Learning proper orthogonal decomposition of complex dynamics using heavy-ball neural ODEs, J. Sci. Comput., Volume 95 (2023) no. 2, 54 | MR
[42] Reduced basis approximations of parameterized dynamical partial differential equations via neural networks, Found. Data Sci., Volume 7 (2025) no. 1, pp. 338-362 Sandia National Laboratories (SNL-NM), report number SAND–2025-04099J | DOI | MR
[43] The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Eng., Volume 139 (1996) no. 1, pp. 289-314 | DOI | MR
[44] On the Spectral Bias of Neural Networks (2019) | arXiv
[45] Proper Generalized Decomposition based dynamic data-driven control of thermal processes, Comput. Methods Appl. Mech. Eng., Volume 213–216 (2012), pp. 29-41 | DOI
[46] A Parsimonious Separated Representation Empowering PINN-PGD-Based Solutions for Parametrized Partial Differential Equations, Mathematics, Volume 12 (2024) no. 15, 2365, 13 pages | DOI
Cité par Sources :
Commentaires - Politique