Dimensionality Reduction and Reduced Order Modeling for Traveling Wave Physics

Abstract

We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and subspace clustering to robustly identify translational invariances that can be leveraged to build improved reduced order models (ROMs). Invariances, whether translational or rotational, are well known to compromise the ability of ROMs to produce accurate and/or low-rank representations of the spatio-temporal dynamics. However, by discovering translations in a principled way, data can be shifted into a coordinate systems where quality, low-dimensional ROMs can be constructed. This approach can be used on either numerical or experimental data with or without knowledge of the governing equations. We demonstrate our method on a variety of PDEs of increasing difficulty, taken from the field of fluid dynamics, showing the efficacy and robustness of the proposed approach.

arXiv comment14 pages, 8 figures

References in corpus (12)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• On Dynamic Mode Decomposition: Theory and Applications
• Turbulence Modeling in the Age of Data
• Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
• Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation
• A Novel Model Order Reduction Approach for Navier-Stokes Equations at High Reynolds Number
• A neural network approach for the blind deconvolution of turbulent flows
• Asymmetric, helical and mirror-symmetric travelling waves in pipe flow
• Generating Random Earthquake Events for PTHA
• Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations
• Hopf fibrations for turbulent pipe flows
• Symmetry reduction of turbulent pipe flows

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• Data-driven Modeling of Rotating Detonation Waves
• Predicting Solar Wind Streams from the Inner-Heliosphere to Earth via Shifted Operator Inference
• Data-Driven Modeling of Nonlinear Traveling Waves
• A Deep Learning Powered Numerical Relativity Surrogate for Binary Black Hole Waveforms
• A robust shifted proper orthogonal decomposition: Proximal methods for decomposing flows with multiple transports
• Learning latent representations in high-dimensional state spaces using polynomial manifold constructions
• Characterization of elastic topological states using dynamic mode decomposition
• Deep Learning-Based Prediction of High Explosive Induced Fluid Dynamics