Non-intrusive reduced order modeling of nonlinear problems using neural networks

By Jan S. Hesthaven, Stefano Ubbiali. Journal of Computational Physics, 2018

632 citations recorded by the source corpus.

Abstract We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approximate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the Latin hypercube sampling (LHS) and the Levenberg–Marquardt (LM) training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method – referred to as POD-NN – suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier–Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4d1e661894f5ac8e56cccf185db307f2ad298bda

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Citations in the corpus

Data Driven Governing Equations Approximation Using Deep Neural Networks (2018, methodology): “Eﬀorts have been devoted to the use of ANN for various aspects of scientiﬁc computing, including construction of reduced order model ([22]), aiding solution of conservation laws ([37]), multiscale problems ([8, 51]), solving and learning systems involving ODEs and PDEs 2.”
POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition (2021, methodology): “…data-driven and less intrusive way (avoiding, e.g., the costs entailed by the projection stage of projection-based ROMs); for instance, in [25, 26, 27, 28, 29] the use of ANNs or Gaussian Processes (GPs) regression models has been proposed to approximate the mapping from the input parameters…”
A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs (2020, methodology): “Moreover, similar trends of suitable indicators of the error between the FOM and the ROM approximation with respect to the reduced dimension can also be observed in [11,29].”
A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime (2019, methodology): “n found either by standard Galerkin projection Rowley et al. (2004); Cui et al. 2 (2015); Afkham and Hesthaven (2017) or regression-based strategies Yu and Hesthaven (2018); Guo and Hesthaven (2018); Hesthaven and Ubbiali (2018). Apart from issues of eciency and stability, both gPC and RB approaches”
Data-driven reduced order modeling for time-dependent problems (2019, methodology): “Furthermore, we would like to refer to [19] for some quantitative comparisons of the online accuracy and efficiency between the conventional RB method and the regression-based RB scheme, and [15] for a comparison of computational times at different stages in the regression-based framework.”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, neutral): “The application of variants of ANNs as regression models for the dynamics of low-order states (e.g., POD amplitudes) has gained substantial popularity, 364–366 owing to the availability of open-source and user-friendly ML libraries.”
Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, neutral): “[298] J.”