Synthetic Data-Guided Symmetric Neural Network Approximation in Banach Spaces

This paper develops a Banach space-valued approximation framework based on symmetrized neural network (SNN) operators generated by a deformation-dependent sigmoidal activation function. Symmetry is introduced directly at the activation level through a reciprocal-deformation mechanism, yielding a positive, even, normalized, and localized density kernel satisfying the partition of unity. The resulting construction provides normalized compact-interval and whole-line quasi-interpolation operators for Banach space-valued functions. Quantitative pointwise and uniform convergence estimates are established through the first modulus of continuity and are extended to higher-order and Caputo–Bochner fractional approximation. Numerical diagnostics support the theoretical kernel properties, and fractional approximation experiments compare the SNN and classical NN operators under common computational conditions. A controlled blind-prediction experiment on a synthetic monthly temperature-like series uses a strict fit–validation–test protocol and a parameter-matched operator comparison, with seasonal ARIMA and MLP models as external baselines. Across five independent realizations, the SNN attains the best mean predictive performance, with R2=0.9500, NMAE =0.0452, and NRMSE =0.0570. A vector-valued experiment in Y=R2 further illustrates the non-scalar applicability of the Banach space framework. In addition, the normalized SNN kernel weights provide an intrinsic node-level interpretation mechanism without requiring an external post hoc explainability method.