Projection-based model-order reduction via graph autoencoders suited for unstructured meshes

Abstract

This paper presents the development of a graph autoencoder architecture capable of performing projection-based model-order reduction (PMOR) using a nonlinear manifold least-squares Petrov-Galerkin (LSPG) projection scheme. The architecture is particularly useful for advection-dominated flows modeled by unstructured meshes, as it provides a robust nonlinear mapping that can be leveraged in a PMOR setting. The presented graph autoencoder is constructed with a two-part process that consists of (1) generating a hierarchy of reduced graphs to emulate the compressive abilities of convolutional neural networks (CNNs) and (2) training a message passing operation at each step in the hierarchy of reduced graphs to emulate the filtering process of a CNN. The resulting framework provides improved flexibility over traditional CNN-based autoencoders because it is readily extendable to unstructured meshes. We provide an analysis of the interpretability of the graph autoencoder's latent state variables, where we find that the Jacobian of the decoder for the proposed graph autoencoder provides interpretable mode shapes akin to traditional proper orthogonal decomposition modes. To highlight the capabilities of the proposed framework, which is named geometric deep least-squares Petrov-Galerkin (GD-LSPG), we benchmark the method on a one-dimensional Burgers' model with a structured mesh and demonstrate the flexibility of GD-LSPG by deploying it on two test cases for two-dimensional Euler equations that use an unstructured mesh. The proposed framework is more flexible than using a traditional CNN-based autoencoder and provides considerable improvement in accuracy for very low-dimensional latent spaces in comparison with traditional affine projections.

42 pages, 22 figures

References in corpus (21)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows
• A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions
• Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders
• Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems
• Deep Fluids: A Generative Network for Parameterized Fluid Simulations
• A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
• Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data
• Machine-learning-based reduced order modeling for unsteady flows around bluff bodies of various shapes
• Molecular graph generation with Graph Neural Networks
• Data-driven reduced-order models via regularized operator inference for a single-injector combustion process
• Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms
• The method of freezing as a new tool for nonlinear reduced basis approximation of parameterized evolution equations
• Operator inference for non-intrusive model reduction with quadratic manifolds
• LaSDI: Parametric Latent Space Dynamics Identification
• A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling
• A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations
• Physics-aware Reduced-order Modeling of Transonic Flow via $\beta$-Variational Autoencoder
• gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification
• Projection-tree reduced order modeling for fast N-body computations
• Multidimensional upwind hydrodynamics on unstructured meshes using Graphics Processing Units I. Two-dimensional uniform meshes