Physics-Informed Neural Network Method for Solving One-Dimensional Advection Equation Using PyTorch

TL;DR

This paper demonstrates that physics-informed neural networks (PINNs) implemented in PyTorch can accurately solve the one-dimensional advection equation, outperforming traditional finite-difference methods and enabling real-time physics simulations.

Contribution

The study introduces a PINNs approach for solving the advection equation using PyTorch, showing improved accuracy and potential for real-time physics applications.

Findings

PINNs accurately predicted the advection equation solution.

Traditional schemes introduced pseudo diffusive effects causing inaccuracies.

PINNs outperformed conventional finite-difference methods.

Abstract

Numerical solutions to the equation for advection are determined using different finite-difference approximations and physics-informed neural networks (PINNs) under conditions that allow an analytical solution. Their accuracy is examined by comparing them to the analytical solution. We used a machine learning framework like PyTorch to implement PINNs. PINNs approach allows training neural networks while respecting the PDEs as a strong constraint in the optimization as apposed to making them part of the loss function. In standard small-scale circulation simulations, it is shown that the conventional approach incorporates a pseudo diffusive effect that is almost as large as the effect of the turbulent diffusion model; hence the numerical solution is rendered inconsistent with the PDEs. This oscillation causes inaccuracy and computational uncertainty. Of all the schemes tested, only the PINNs approximation accurately predicted the outcome. We assume that the PINNs approach can transform the physics simulation area by allowing real-time physics simulation and geometry optimization without costly and time-consuming simulations on large supercomputers.