Approximation Capabilities of Feedforward Neural Networks with GELU Activations

2025-12-25

AI Summary

This work investigates the joint uniform approximation capability of GELU-activated feedforward neural networks for multivariate polynomials, exponential functions, and reciprocal functions—along with their arbitrary-order derivatives. Addressing the lack of theoretical frameworks that simultaneously control high-order derivatives, we establish, for the first time, explicit uniform error bounds for both the target function and all its prescribed-order derivatives. We propose a constructive multiplicative approximation scheme and systematically extend it to division and exponentiation, ensuring global boundedness of all derivative orders. Through layerwise derivative control and asymptotic analysis at infinity, we derive explicit upper bounds on network width, weight magnitudes, and characterize the network’s behavior at infinity. Our results show that, for these fundamental function classes, the uniform approximation error in derivatives decays exponentially with network width.

Technology Category

Machine Learning: Mixture of Experts (MoE)Search and Optimization: Learning to SearchNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSocial Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

We derive an approximation error bound that holds simultaneously for a function and all its derivatives up to any prescribed order. The bounds apply to elementary functions, including multivariate polynomials, the exponential function, and the reciprocal function, and are obtained using feedforward neural networks with the Gaussian Error Linear Unit (GELU) activation. In addition, we report the network size, weight magnitudes, and behavior at infinity. Our analysis begins with a constructive approximation of multiplication, where we prove the simultaneous validity of error bounds over domains of increasing size for a given approximator. Leveraging this result, we obtain approximation guarantees for division and the exponential function, ensuring that all higher-order derivatives of the resulting approximators remain globally bounded.

Problem

• Bounding approximation errors for functions and derivatives
• Analyzing GELU neural networks for elementary function approximation
• Ensuring global boundedness of higher-order derivative approximators

Innovation

• GELU activation for simultaneous function-derivative approximation
• Constructive multiplication approximation with increasing domain validity
• Guaranteed bounded higher-order derivatives for division and exponential