To fulfill future diverse user requirements, 6G networks are envisioned to provide everyone-centric customized services ubiquitously and precisely. However, the diversity in user requirements and the heterogeneity in network resources challenge conventional network operators in network management and service provision. In this paper, we investigate the artificial intelligence (AI) service provision in the multi-layer heterogeneous network. To provide ubiquitous intelligence to users with different computing requirements, an intelligence-native network architecture is designed. Based on the proposed architecture and the AI model stitching mechanism, we formulate the joint AI provision and access selection problem as a mixed integer non-linear programming (MINLP) problem to maximize the average user satisfaction value and user satisfaction rate. Then, a heuristic solution based on Dung Beetle algorithm is proposed to optimize the AI model selection, AI service deployment, user access, and stitching coefficient jointly. Extensive simulations are conducted to evaluate the performance of our proposed architecture and algorithm.