On approximation by neural networks in Lp spaces

The directly constructed neural network operators have excellent performances in the theoretical verification of generalization ability, reduction of the training time of neural networks, and avoidance of the local minimum solution in the optimization process of neural networks. This paper introduces a new class of Durrmeyer-type neural network operators and investigates their approximation properties in the [Formula: see text] space for [Formula: see text]. The main contributions are threefold: (1) The introduction of truncated absolute discrete moments and truncated absolute integral moments to characterize activation functions; (2) The construction of a new neural network operator via a specialized function extension technique; (3) The establishment of direct and inverse approximation theorems. The direct theorems are proved by exploiting the equivalence between the [Formula: see text]-functional and the modulus of continuity of the target function, while the inverse theorems are derived by estimating the derivatives of the operators and applying the Berens–Lorentz lemma. Combining the direct and inverse results yields the essential approximation rate. The theoretical findings are applied to specific activation functions, including the Jackson kernel, [Formula: see text]-deformed hyperbolic tangent, and B-splines, with supporting numerical validation.