Session: 1.2 - Machine Learning, Reduced Order Modeling in CFD and Design Optimization

Paper Number: 158656

158656 - Hybrid Autoencoder/Galerkin Approach for Nonlinear Reduced Order Modelling 

Abstract: 

This study introduces a nonlinear Reduced Order Model (ROM) for fluid dynamics, combining Proper
Orthogonal Decomposition (POD) with deep learning error correction. Our approach merges the in-
terpretability and physical adherence of POD Galerkin ROMs with the predictive capabilities of deep
learning. The hybrid model addresses errors within and outside the POD subspace. Firstly, POD gener-
ates part of the reduced state, complemented by an autoencoder compressing only the unretained POD
modes. Thus, the most energetic modes are computed interpretably, while the least energetic are han-
dled with a superior reduction method. Secondly, the time integration employs a hybrid neural Ordinary
Differential Equation (neural ODE) [1]. A POD ROM estimates part of the dynamics, and a deep learn-
ing model corrects its error. Using Neural ODE aligns the model with underlying physics for enhanced
stability and accuracy.
The proposed method differs from current hybrid methods [2] operating solely in the POD subspace
and using Mori-Zwanzig time dependency [3], posing potential initialisation issues.
Experiments were conducted on the viscous Burgers’ equation and the parametric circular cylinder
flow, and we are working on the Kuramoto–Sivashinsky equation and the fluidic pinball. Accuracy and
numerical complexity are compared to classical POD Galerkin ROMs, fully data-driven models, and
concurrent hybrid methods.

References
[1] R.T.Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, “Neural ordinary differential equa-
tions,” NeurIPS,2018.
[2] 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.
[3] R. Zwanzig, Nonequilibrium Statistical Mechanics. Oxford University Press., 2000.

Presenting Author: Nicolas Lepage M2N-CNAM

Presenting Author Biography: