Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

•Deep auto-regressive dense encoder-decoder surrogate for predicting transient PDEs.•Physics-constrained learning enables the model to learn dynamics without training data.•A Bayesian framework is proposed for interpretable uncertainty quantification of the models' predictions at each time-step...

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Bibliographic Details
Published in:Journal of computational physics Vol. 403; p. 109056
Main Authors: Geneva, Nicholas, Zabaras, Nicholas
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.02.2020
Elsevier Science Ltd
Subjects:
Artificial neural networks
Auto-regressive model
Burgers equation
Coders
Computational physics
Convolutional encoder-decoder
Deep learning
Deep neural networks
Dynamic partial differential equations
Encoders-Decoders
Machine learning
Mathematical models
Model testing
Nonlinear systems
Numerical analysis
Numerical methods
Partial differential equations
Physics-informed machine learning
Regression analysis
Solvers
Training
Uncertainty
Uncertainty quantification
Physics-informed machine learning
Uncertainty quantification
Auto-regressive model
Deep neural networks
Dynamic partial differential equations
Convolutional encoder-decoder
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:•Deep auto-regressive dense encoder-decoder surrogate for predicting transient PDEs.•Physics-constrained learning enables the model to learn dynamics without training data.•A Bayesian framework is proposed for interpretable uncertainty quantification of the models' predictions at each time-step.•The auto-regressive model is tested on several non-linear PDE systems with features including turbulence and shock discontinuities. In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2019.109056