Reduced Order Modeling Using Advection-Aware Autoencoders

Abstract

Physical systems governed by advection-dominated partial differential equations (PDEs) are found in applications ranging from engineering design to weather forecasting. They are known to pose severe challenges to both projection-based and non-intrusive reduced order modeling, especially when linear subspace approximations are used. In this work, we develop an advection-aware (AA) autoencoder network that can address some of these limitations by learning efficient, physics-informed, nonlinear embeddings of the high-fidelity system snapshots. A fully non-intrusive reduced order model is developed by mapping the high-fidelity snapshots to a latent space defined by an AA autoencoder, followed by learning the latent space dynamics using a long-short-term memory (LSTM) network. This framework is also extended to parametric problems by explicitly incorporating parameter information into both the high-fidelity snapshots and the encoded latent space. Numerical results obtained with parametric linear and nonlinear advection problems indicate that the proposed framework can reproduce the dominant flow features even for unseen parameter values.

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

“…with: u(x, 0) = x 1+exp( Re 16 (4x 2 −1)) and u(0, t) = u(1, t) = 0. An analytical solution for the velocity field of the convection-diffusion problem described above is given by [29]:…”

“…12b. The downstream part of the dam is considered as dry whereas the free surface of the upstream part is considered as a random input parameter whose values are uniformly generated within its plausible variability range η up ∈ U [29,32] m. The snapshot matrix is obtained by running the numerical solver for each value of the upstream free surface, selected randomly from the generated sample set, for the whole N t = 100 simulation time steps that constitute the temporal domain (t ∈ [0, 50] s). For each parameter-time combination, a so-called high-fidelity solution is stored in a vector of dimension of N x = 10 200 representing the free surface values at each node of the computational domain.…”

“…CAEs offer the possibility of sharing coefficients and local patch connection allowing a significant reduction in the trainable parameters. Some configurations combine CAEs with LSTM and TCN [28,29] to provide a tool for surrogate modeling predictions, particularly for time predictions.…”

Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks

“…Nevertheless, in all cases the network is hard to train at the presence of traveling features, such as shock, fronts, and gradients. Some of the successful remedies in [7,10,11] can be explained by breaking the Kolmogorov n-width, a recognized paradigm in finite element method of solving PDEs [29,30], data assimilation [31], NNs-based reduced order models (ROMs) [32][33][34], projection-based ROMs [30,[35][36][37][38][39][40][41][42][43], flexDeepONet [44], and projection-based ROMs on NNbased manifolds [45,46]. To demonstrate the effect of the proposed remedies on the rate of decay of (normalized) singular values, we consider the synthetic data of fig.…”

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

“…Various data-driven ML-based frameworks have been proposed to model the propagation of system dynamics in latent space. Some of the more highly successful examples involve the use of deep neural networks (DNNs) [15], long-short-term memory (LSTM) networks [16,17,18,19] , neural ordinary differential equations (NODE) [20,21,19], and temporal convolutional networks (TCNs) [22,23].…”