Deep Galerkin Method for Burgers’ Equation: A Comparative Study with Exact Solutions
DOI:
https://doi.org/10.65904/3083-3590.2025.01.03Keywords:
Deep Galerkin Method (DGM), Burgers Equation, Physics-Informed Neural Networks (PINNs), Nonlinear Partial Differential Equations, Mesh-Free Methods, AMS MSC: 68T07, 35Q35, 68T07, 35-XX, 65M75Abstract
This study evaluates the performance of the Deep Galerkin Method (DGM) for solving the Burgers equation in both one and two spatial dimensions. Through extensive numerical experiments, we show that DGM accurately reproduces the exact analytical solutions, effectively capturing the dynamics of viscous shock waves with errors below . The method demonstrates fast convergence, inherent geometric flexibility, and avoids the limitations of traditional mesh-based schemes, positioning DGM as a robust and efficient mesh-free approach for solving nonlinear partial differential equations.
References
B. Damski, Formation of shock waves in a Bose-Einstein condensate. Physical Review A-Atomic, Molecular, and Optical Physics, 69(4), 043610. (2004). https://doi.org/10.1103/PhysRevA.69.043610
W. D. McComb, The physics of fluid turbulence. Oxford University Press, (1990). https://doi.org/10.1093/oso/9780198561606.001.0001
S. T. Rachev, L. Rüschendorf, Mass Transportation Problems: Volume I: Theory. Springer, New York, (1998).
A. D. May, Traffic flow fundamentals. Prentice Hall, (1990).
J. Christensen, L. Martin-Moreno, F. J. Garcia-Vidal, Theory of resonant acoustic transmission through subwavelength apertures. Physical Review Letters, 101(1), 014301 (2008). https://doi.org/10.1103/PhysRevLett.101.014301
A. Bressan, K. T. Nguyen, Singularity formation for the fractional Burgers equation. Archive for Rational Mechanics and Analysis, 239(1), 1-21 (2021).
L. Chevillard, R. Robert, V. Vargas, et al., Unified multifractal description of the increments of the solutions of the inviscid and viscous Burgers equation. Physical Review E, 99(3), 033107 (2019).
A. Bandopadhyay, T. Le Borgne, Mixing and reaction kinetics in porous media: An experiment-inspired pore-scale description. Physical Review Fluids, 2(1), 012101 (2017).
A. K. Gupta, P. Redhu, Jamming transitions and the effect of interruption probability in a lattice hydrodynamic model with passing. Physica A: Statistical Mechanics and its Applications, 572, 125904 (2021).
N. Sugimoto, Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. Journal of Fluid Mechanics, 825, 1-17 (2017).
A. Khodadadian, S. Jamali, A. Mohebbi, M. Dehghan, A hybrid PINN-boundary layer method for viscous Burgers equation. Mathematics, 12(21), 3430 (2024). https://doi.org/10.3390/math12213430
R. K. Jena, S. Singh, S. Sahoo, Analytical solutions of generalized fractional Burgers equation using homotopy perturbation method. Symmetry, 15(3), 634 (2023). https://doi.org/10.3390/sym15030634
T. S. Khan, G. Akram, A deep Galerkin method for generalized Burgers-Fisher equations. Journal of Applied and Computational Mechanics, 9(4), 823-836 (2023).
E. Hopf, The partial differential equation . Communications on Pure and Applied Mathematics, 3(3), 201-230 (1950). https://doi.org/10.1002/cpa.3160030302
J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 9(3), 225-236 (1951). https://doi.org/10.1090/qam/42889
