This study presents a novel framework for precise force control of fin-actuated underwater robots by integrating a deep neural network (DNN)-based surrogate model with reinforcement learning (RL). To address the complex interactions with the underwater environment and the high experimental costs, a DNN surrogate model acts as a simulator for enabling efficient training for the RL agent. Additionally, grid-switching control is applied to select optimized models for specific force reference ranges, improving control accuracy and stability. Experimental results show that the RL agent, trained in the surrogate simulation, generates complex thrust motions and achieves precise control of a real soft fin actuator. This approach provides an efficient control solution for fin-actuated robots in challenging underwater environments.