Reduced-order modeling for stochastic large-scale and time-dependent problems using deep spatial and temporal convolutional autoencoders

Abstract

A non-intrusive reduced order model based on convolutional autoencoders (NIROM-CAEs) is proposed as a data-driven tool to build an efficient nonlinear reduced-order model for stochastic spatio-temporal large-scale physical problems. The method uses two 1d-convolutional autoencoders (CAEs) to reduce the spatial and temporal dimensions from a set of high-fidelity snapshots collected from the high-fidelity numerical solver. The encoded latent vectors, generated from two compression levels, are then mapped to the input parameters using a regression-based multilayer perceptron (MLP). The accuracy of the proposed approach is compared to that of the linear reduced-order technique-based artificial neural network (POD-ANN) through two benchmark tests (the one-dimensional Burgers and Stoker's solutions) and a hypothetical dam-break flow problem, with an unstructured mesh and over a complex bathymetry river. The numerical results show that the proposed nonlinear framework presents strong predictive abilities to accurately approximate the statistical moments of the outputs for complex stochastic large-scale and time-dependent problems, with low computational cost during the predictive online stage.

Deep convolutional architectures for extrapolative forecasts in time-dependent flow problems

“…The restriction (dim(z i ) = m) (n = dim(v i )) forces the autoencoder model to learn the salient features of the input data via compression into a low-dimensional space and to then reconstruct the input, instead of directly learning the identity function. Autoencoder architectures are generally comprised of MLPs (called AAs) [18], convolutional neural network autoencoders (called CAEs) [23,25,28], or a combination of both. While smallsized problems can be effectively modeled via an MLP architecture, problems involving data of high spatial complexity require CAE autoencoders for effective and accelerated spatial compression.…”

“…, in which c m = gh m [28]. For each selected value in the generated sample set of the upstream water level, the analytical solution given above is evaluated over 1000 nodes (n s = 1000) that contain the computational domain x ∈ [0, 100] m for all 450 time-steps (T = 450) of the temporal domain t ∈ [0, 3.6]s. Four-hundred solution vectors, at random time steps, train the autoencoder network, and the remaining 50 vectors are used for the validation.…”

“…Wu et al [22] developed a POD and TCN-based neural network for making predictions on the viscous periodic flow past a cylinder case. Abdedou et al [28] proposed two CAE architectures to compress the high-dimensional snapshot matrices obtained from numerical solvers for the Burgers', Stoker's, and shallow-water equations in space and time and performed parameterization on the compressed latent space. Jacquier et al [29] employed uncertainty quantification methods-Deep Ensembles and Variational Inference-based Bayesian Neural Networks on the POD-ANN order-reduction method to perform predictions within and outside of the training domain on problems such as shallow water equations for flood prediction, and generated probabilistic flooding maps aware of model uncertainty.…”

Deep Convolutional Architectures for Extrapolative Forecasts in Time-dependent Flow Problems

“…) forces the autoencoder model to learn the salient features of the input data via compression into a low-dimensional space and to then reconstruct the input, instead of directly learning the identity function. Autoencoder architectures are generally comprised of MLPs (called AAs) [18], convolutional neural network autoencoders (called CAEs) [25,23,28], or a combination of both. While small-sized problems can be effectively modelled via an MLP architecture, problems involving data of high spatial complexity require CAE autoencoders for effective and accelerated spatial compression.…”

“…Wu et al [22] developed a POD and TCN-based neural network for making predictions on the viscous periodic flow past a cylinder case. Abdedou et.al [28] proposed two CAE architectures to compress the high-dimensional snapshot matrices obtained from numerical solvers for the Burgers', Stoker's, and shallow-water equations in space and time and performed parameterization on the compressed latent space. Jacquier et al [29] employed uncertainty quantification methods -Deep Ensembles and Variational Inference-based Bayesian Neural Networks on the POD-ANN orderreduction method to perform predictions within and outside of the training domain on problems such as shallow water equations for flood prediction, and generated probabilistic flooding maps aware of model uncertainty.…”