Efficient nonlinear manifold reduced order model

Abstract

Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, such as advection-dominated flow phenomena, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed an efficient nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models (FOMs). The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 2D Burgers' equations with a high Reynolds number. A speed-up of up to 11.7 for 2D Burgers' equations is achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique.

10 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:2009.11990

References in corpus (14)

• Adam: A Method for Stochastic Optimization
• DeepXDE: A deep learning library for solving differential equations
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
• POD/DEIM Nonlinear model order reduction of an ADI implicit shallow water equations model
• Adaptive $h$-refinement for reduced-order models
• Conservative model reduction for finite-volume models
• Generating Random Earthquake Events for PTHA
• POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification
• Accelerating design optimization using reduced order models

Cited by in corpus (14)

• LaSDI: Parametric Latent Space Dynamics Identification
• Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques
• Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction
• Component-wise reduced order model lattice-type structure design
• Reduced order models for Lagrangian hydrodynamics
• Nonlinear proper orthogonal decomposition for convection-dominated flows
• GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder
• A fast and accurate domain-decomposition nonlinear manifold reduced order model
• Mapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models
• Registration-based model reduction of parameterized two-dimensional conservation laws
• Gappy AE: A Nonlinear Approach for Gappy Data Reconstruction using Auto-Encoder
• Registration-based model reduction in complex two-dimensional geometries
• A FOM/ROM Hybrid Approach for Accelerating Numerical Simulations
• Learning reduced order models from data for hyperbolic PDEs