Ablation Study of Gaussian RBF-KAN for PDEs Approximation

Kolmogorov-Arnold Networks replace the fixed activation of multilayer perceptrons with learnable univariate functions assigned to the edges of the computational graph. The shared-basis Gaussian variant of this architecture parameterizes the radial basis functions globally per layer and yields a substantial reduction of the parameter count while preserving the convergence properties of classical radial basis approximation. The accuracy of the resulting model on partial differential equation benchmarks depends on a small set of discretization hyperparameters: the cardinality of the Gaussian basis, the depth of the network, and the width of each hidden layer. The interaction of these factors has not yet been quantified systematically. The present work addresses this gap through a controlled ablation on two canonical benchmarks of distinct mathematical character, such as hyperbolic and elliptic partial differential equations. The results identify a stable regime of small bases and shallow networks in which the convergence rate predicted by classical approximation theory is realized in practice, and a high-capacity regime in which over-parameterization degrades accuracy through high-frequency residual artifacts rather than through classical overfitting. The pattern is consistent across the two equation classes, with a residual geometry that reflects the underlying differential operator. The findings provide concrete prescriptions for the deployment of the architecture in physicsinformed neural network pipelines.