On the Performance of Physics-Based Neural Networks for Symmetric and Asymmetric Domains: A Comparative Study and Hyperparameter Analysis

This work investigates the use of physics-informed neural networks (PINNs) for solving representative classes of differential and integro-differential equations, including the Burgers, Poisson, and Volterra equations. The examples presented are chosen to address both symmetric and asymmetric domains. PINNs integrate prior physical knowledge with the approximation capabilities of neural networks, allowing the modeling of physical phenomena without explicit domain discretization. In addition to evaluating accuracy against analytical solutions (where available) and established numerical methods, the study systematically examines the impact of key hyperparameters—such as the number of hidden layers, neurons per layer, and training points—on solution quality and stability. The impact of a symmetric domain on solution speed is also analyzed. The experimental results highlight the strengths and limitations of PINNs and provide practical guidelines for their effective application as an alternative or complement to traditional computational approaches.