A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

Abstract

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes.

Explicit Numerical Manifold Characteristic Galerkin Method for Solving Burgers’ Equation

“…Based on the one-dimensional dual-cover system, our method adopts a PUM-like technique to connect PCs, and there is no requirement for MC mesh shapes. (6) Various initial-boundary value problems are considered in Section 6, including trigonometric functions, polynomial functions, particular traveling solutions and the Riemann problem with two constant initial values. Numerical evidence underscores that the results of the ENMCGM are in concordance with analytical solutions and the published results for the Burgers' equation, particularly for high Reynolds numbers.…”

“…Much effort has been dedicated to designing accurate and stable computing schemes to solve Burger's equation for large Reynolds numbers. According to the manner for spacial discretization, these schemes can be roughly categorized into mesh-based approaches [2][3][4][5][6][7][8] and mesh-free methods [9,10]. However, mesh-based approaches, such as finite difference/volume/element methods, seriously rely on deliberated meshes to discretize problem domain onto grid lines or elements to acquire accurate spacial approximation.…”

Prediction and Optimization of Thermal Loads in Buildings with Different Shapes by Neural Networks and Recent Finite Difference Methods

“…This study showed how wall design and orientation can optimize the thermal comfort of a building. Our next work will simulate the windows and the roof and we will use ANNs to solve the transient PDE itself, similarly to the work [43].…”

Displacement Reconstruction Based on Physics-Informed DeepONet Regularizing Geometric Differential Equations of Beam or Plate

“…Guided by physical loss terms, PINN requires fewer or no paired input-output observations to be trained and provides better generalization. The application of PINN has expanded to various fields, including fluid mechanics [20], biomedicine [21], materials science [22], fracture mechanics [23], power systems [24], and scientific machine learning (SciML) [25]. Therefore, PINN is a highly promising technique for addressing the issue of displacement reconstruction.…”