A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs

By Stefania Fresca, Luca Dede', Andrea Manzoni. Journal of Scientific Computing, 2020

348 citations recorded by the source corpus.

Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/05e340e290bdc236249f2be0c7749f6c9bebc483

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Electrical and Electronic Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

Model Reduction and Neural Networks for Parametric PDEs (2020, methodology): “Furthermore the works [50, 28, 31, 28] proposed a model reduction approach for dynamical systems by use of dimension reducing neural networks (autoencoders).”
POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition (2021, methodology): “Regarding the setting of the optimization algorithm, the way to select the hyperparameters and the architecture of the neural networks, we refer to [9, 10].”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, positive): “The introduction of neural network regression into ROM was highly motivated by the desire to construct purely data-driven nonin-trusive ROM (NIROM) frameworks, 8,360 which solely rely on data to learn the dynamics of the relevant solution manifold without the need to access the governing equations.”