Massive computational resources are required by a booming number of artificial intelligence (AI) services in the communication network of the smart grid. To alleviate the computational pressure on data centers, edge computing first network (ECFN) can serve as an effective solution to realize distributed model training based on data parallelism for AI services in smart grid. Due to AI services with diversified types, an edge data center has a changing workload in different time periods. Selfish edge data centers from different edge suppliers are reluctant to share their computing resources without a rule for fair competition. AI services-oriented dynamic computational resource scheduling of edge data centers affects both the economic profit of AI service providers and computational resource utilization. This letter mainly discusses the partition and distribution of AI data based on distributed model training and dynamic computational resource scheduling problems among multiple edge data centers for AI services. To this end, a mixed integer linear programming (MILP) model and a Deep Reinforcement Learning (DRL)-based algorithm are proposed. Simulation results show that the proposed DRL-based algorithm outperforms the benchmark in terms of profit of AI service provider, backlog of distributed model training tasks, running time and multi-objective optimization.