Stable and Efficient Gaussian-Based Kolmogorov–Arnold Networks

Kolmogorov–Arnold Networks employ learnable univariate activation functions on edges rather than fixed node nonlinearities. Standard B-spline implementations require O(3KW) parameters per layer (K basis functions, W connections). We introduce shared Gaussian radial basis functions with learnable centers μk(l) and widths σk(l) maintained globally per layer, reducing parameter complexity to O(KW+2LK) for L layers—a threefold reduction, while preserving Sobolev convergence rates O(hs−Ω). Width clamping at σmin=10−6 and tripartite regularization ensure numerical stability. On MNIST with architecture [784,128,10] and K=5, RBF-KAN achieves 87.8% test accuracy versus 89.1% for B-spline KAN with 1.4× speedup and 33% memory reduction, though generalization gap increases from 1.1% to 2.7% due to global Gaussian support. Physics-informed neural networks demonstrate substantial improvements on partial differential equations: elliptic problems exhibit a 45× reduction in PDE residual and maximum pointwise error, decreasing from 1.32 to 0.18; parabolic problems achieve a 2.1× accuracy gain; hyperbolic wave equations show a 19.3× improvement in maximum error and a 6.25× reduction in L2 norm. Superior hyperbolic performance derives from infinite differentiability of Gaussian bases, enabling accurate high-order derivatives without polynomial dissipation. Ablation studies confirm that coefficient regularization reduces mean error by 40%, while center diversity prevents basis collapse. Optimal basis count K∈[3,5] balances expressiveness and overfitting. The architecture establishes Gaussian RBFs as efficient alternatives to B-splines for learnable activation networks with advantages in scientific computing.