Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks

06/13/2019
by   Nicholas Geneva, et al.
0

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 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 transient systems with non-linear dynamics 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.

READ FULL TEXT

page 24

page 29

page 32

page 36

page 38

page 39

page 41

page 42

research
09/13/2020

Optimal Bounds on Nonlinear Partial Differential Equations in Model Certification, Validation, and Experimental Design

We demonstrate that the recently developed Optimal Uncertainty Quantific...
research
03/12/2021

Physics-Informed Deep-Learning for Scientific Computing

Physics-Informed Neural Networks (PINN) are neural networks that encode ...
research
01/18/2019

Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

Surrogate modeling and uncertainty quantification tasks for PDE systems ...
research
12/14/2022

Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are gaining popularity as a met...
research
05/22/2023

Uncertainty and Structure in Neural Ordinary Differential Equations

Neural ordinary differential equations (ODEs) are an emerging class of d...
research
08/05/2021

Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data

Scientific machine learning has been successfully applied to inverse pro...
research
04/13/2022

Stochastic Finite Volume Method for Uncertainty Quantification of Transient Flow in Gas Pipeline Networks

We develop a weakly intrusive framework to simulate the propagation of u...

Please sign up or login with your details

Forgot password? Click here to reset