Construction and approximation rate of neural network operators for Broad Learning System

This paper investigates the construction and approximation properties of neural network operators for Broad Learning Systems (BLS) based on general activation functions. By introducing a function extension technique, our constructed operators require only information of the target function on a compact interval, significantly relaxing the stringent requirements on the domain and smoothness of the target functions in existing theories. Under appropriate asymptotic conditions on the activation functions, we establish a comprehensive theoretical framework for approximation. Firstly, for functions with fractional or integer-order smoothness, we derive sharp upper bounds for the approximation error (direct theorems). Secondly, provided the activation function is differentiable with a certain decay rate, we prove an inverse theorem, revealing the intrinsic relationship between the approximation rate and the smoothness of the target function. A corollary on the essential order of approximation is consequently obtained. Our theoretical analysis demonstrates that the constructed BLS operators possess universal approximation capabilities, and their network complexity can be explicitly characterized by the smoothness of the target function. Numerical experiments are provided to validate the theoretical findings. This study provides a solid mathematical foundation for Broad Learning Systems and offers theoretical guidance for their applications.