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

## TL;DR

This paper introduces a physics-constrained deep auto-regressive neural network that models nonlinear PDE systems efficiently, requiring minimal training data and providing uncertainty quantification, demonstrated on complex dynamical systems.

## Contribution

It presents a novel auto-regressive neural network architecture with a Bayesian framework for modeling PDE systems without training data, reducing computational costs significantly.

## Key findings

- Accurately models nonlinear PDE systems like Kuramoto-Sivashinsky and Burgers' equations.
- Provides uncertainty quantification for predictions at each time step.
- Outperforms traditional numerical methods in efficiency and data requirements.

## Abstract

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.