Activation-Induced Symmetric Kernels for Neural Network Approximation with Quantitative Error Analysis

This paper studies symmetrized neural network (SNN) operators generated by an adjustable half-hyperbolic tangent activation function. The construction is based on the paired density kernels ℵt and ℵ1/t, whose average defines the symmetric kernel F. This kernel is positive, even, normalized, and preserves the partition of unity. Using F, we define finite-interval and whole-line SNN operators in the Banach space-valued setting. Pointwise and uniform convergence estimates are obtained through the first modulus of continuity. Higher-order and fractional approximation estimates are also derived, the latter using Caputo–Bochner fractional derivatives. The numerical part compares the nonsymmetrized operator Ln and the symmetrized operator Lns. For n=80, the uniform error decreases from 0.010463 to 0.003479, the root mean square error (RMSE) decreases from 0.006530 to 0.000826, and the coefficient of determination (R2) improves from 0.999675 to 0.999995. This improvement is accompanied by an increase in central processing unit (CPU) time from 0.014161 s to 0.027088 s. The parameter tests further show that the performance depends on the joint choice of n, t, and ξ. Overall, the results indicate that symmetrization improves approximation accuracy, while parameter tuning remains necessary.