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.