ECCOMAS 2024

Hybrid Autoencoder/Galerkin approach for nonlinear reduced order modelling

  • Lepage, Nicolas (CNAM)
  • Beneddine, Samir (ONERA)
  • Fiorini, Camilla (CNAM)
  • Mortazavi, Iraj (CNAM)
  • Sipp, Denis (ONERA)
  • Thome, Nicolas (Sorbonne University)

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This study presents a new nonlinear Reduced Order Model, combining Proper Orthogonal Decomposition (POD) with deep learning residual error correction. Our hybrid model combines the interpretability and adherence to physical principles of POD Galerkin ROMs with the enhanced representation capacity of deep learning. Importantly the hybrid architecture addresses errors within and outside the POD subspace. Firstly, in dimensionality reduction, classical POD is complemented with an autoencoder that compresses only the unretained information from the POD. This enables linear reduction of the most energetic modes, while the least energetic are handled using a more representative method. Secondly, for time integration, a POD ROM predicts part of the dynamics interpretably, and a deep learning model using the Neural ODE [1] framework corrects its error. To our knowledge, the proposed model operates distinctly to existing hybrid models, they employ Mori-Zwanzig-inspired time-dependency [2,3] , correcting the dynamics only within the POD subspace and causing potential initialization issues. This approach was successfully tested on the Burgers and Kuramoto–Sivashinsky (KS) equations and is being extended to the cylinder flow and fluidic pinball test cases. Optimal compositions for the reduced subspace were tested, improving our understanding of the hybrid method's behaviour. This model outperforms some POD methods, fully data-driven methods, and previously mentioned hybrid models. Importantly, our model maintains low computational overhead compared to classical POD-based ROMs, making it particularly attractive for applications in complex fluid systems, with a significant speed-up. [1] R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, “Neural ordinary differential equations” NeurIPS, 2018. [2]. Wang, N. Ripamonti, and J. S. Hesthaven, “Recurrent neural network closure of parametric pod-galerkin reduced-order models based on the mori-zwanzig formalism,” Journal of Computational Physics, vol. 410, p. 109402, 2020. [3] E. Menier, M. A. Bucci, M. Yagoubi, L. Mathelin, and M. Schoenauer, “Cd-rom: Complemented deep reduced order model” Computer Methods in Applied Mechanics and Engineering, vol. 410, p. 115985, 2023.