Evaluation of Sampling Strategies and Physics-Informed Kolmogorov--Arnold Networks in Unbounded Domains
Physics-informed neural networks (PINNs) have emerged as an effective approach for solving partial differential equations (PDEs) by incorporating physical laws into the learning process. However, their application to infinite and semi-infinite domains remains challenging due to the difficulty of representing unbounded regions with a finite number of training points. This work evaluates physics-informed learning strategies for inverse PDE problems in unbounded domains, focusing on the influence of sampling strategies based on uniform, Gaussian, and exponential distributions, and on the use of conventional multilayer perceptrons (MLPs) and Kolmogorov--Arnold Networks (KANs) within the PINN framework. The proposed benchmark considers manufactured inverse problems on infinite and semi-infinite domains, enabling quantitative assessment of reconstruction accuracy and computational efficiency. Results show that Gaussian and exponential sampling, controlled through explicit distribution parameters, improve reconstruction accuracy in the far field and offer a direct mechanism for embedding problem-specific prior knowledge into training. PIKANs, whose edge-based functional representation is grounded in the Kolmogorov--Arnold representation theorem, match or exceed the accuracy of MLP-based PINNs while requiring fewer neurons, though their spline-based formulation increases the computational cost of training, inference, and derivative evaluation.