A Benchmarking Study for Algorithm Selection in Scientific Machine Learning (SciML): PINN vs. gPINN for Solving Partial Differential Equations

Recent advances in physics-informed neural networks (PINN) have highlighted the need for systematic criteria for selecting appropriate algorithms to solve differential equations. This paper presents a numerical comparison between standard PINNs and gradient-enhanced PINNs (gPINNs) used to solve a high-order partial differential equations (PDE). To verify the accuracy and convergence behavior of all the methods, we solve a fourth-order PDE whose analytical solution is known. gPINN is recommended for problems requiring high accuracy in gradient fields or operating with sparse data, whereas standard PINN is advised for strongly nonlinear or computationally constrained scenarios. We synthesize our findings into a practical selection guide; gPINN is recommended for problems requiring high accuracy in gradient fields or operating with sparse data, whereas standard PINN is advised for strongly nonlinear or computationally constrained scenarios. This framework provides a clear, evidence-based policy for algorithm choice in SciML. Beyond numerical comparison, we provide an analytical interpretation linking solver performance to the spectral and stiffness properties of each PDE class, offering a principled basis for algorithm selection.