Physics-Constrained Auto-Regressive Convolutional Neural Networks

Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks [JCP][ArXiv]

Nicholas Geneva, Nicholas Zabaras

Highlights	Bayesian 2D Coupled Burgers' Prediction
Zero Training Data Surrogate Modeling Deep Convolutional Networks Non-linear Dynamics Chaotic Systems Bayesian Framework 500x Faster than FEM

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.

Index

Each of the PDE systems used in the paper is designated its own folder where more information can be found regarding model training, testing and figure replication.

• 1D-KS-SWAG: The 1D Kuramoto-Sivashinsky system.
• 1D-Burger-SWAG: The 1D viscous Burgers' system.
• 2D-Burgers-SWAG: The 2D coupled Burgers' system.

Deep Turbulence Generation

Core Dependencies

• Python 3.6.5
• PyTorch 1.1.0
• Fenics 2019.1.0
• Matplotlib 3.1.1

Plus additional standard packages such as Numpy and Scipy

Note: PyTorch <= 1.2.0 must be used due to a change padding_mode='circular' behavior, for higher PyTorch versions reduce padding to 1 (currently 2 due to a bug in PyTorch versions 1.1.0 and 1.2.0).

Citation

Find this useful or like this work? Cite us with:

@article{geneva2019modeling,
  title = {Modeling the dynamics of {PDE} systems with physics-constrained deep auto-regressive networks},
  journal = {Journal of Computational Physics},
  pages = {109056},
  year = {2019},
  issn = {0021-9991},
  doi = {10.1016/j.jcp.2019.109056},
  url = {http://www.sciencedirect.com/science/article/pii/S0021999119307612},
  author = {Nicholas Geneva and Nicholas Zabaras}
}