PINNs Algorithmic Framework for Simulation of Nonlinear Burgers' Type Models

TL;DR

This paper presents a PINNs-based algorithm for simulating nonlinear Burgers' models, demonstrating its accuracy, flexibility, and potential as a reliable method for solving complex PDEs in 1D and 2D.

Contribution

It introduces a neural network framework tailored for Burgers' equations, including architecture, loss functions, and training, validated through multiple test problems.

Findings

PINNs accurately replicate nonlinear PDE solutions

The method shows competitive accuracy and flexibility

Validated on five diverse Burgers' model problems

Abstract

In this work, a physics-informed neural networks (PINNs) based algorithm is used for simulation of nonlinear 1D and 2D Burgers' type models. This scheme relies on a neural network built to approximate the problem solution and use a trial function that meets the initial data and boundary criteria. First of all, a brief mathematical formulation of the problem and the structure of PINNs, including the neural network architecture, loss construction, and training methodology is described. Finally, the algorithm is demonstrated with five test problems involving variations of the 1D coupled, 2D single and 2D coupled Burgers' models. We compare the PINN-based solutions with exact results to assess accuracy and convergence of the developed algorithm. The results demonstrate that PINNs may faithfully replicate nonlinear PDE solutions and offer competitive performance in terms of inaccuracy and flexibility. This work demonstrates the potential of PINNs as a reliable approach to solving complex time-dependent PDEs.