Shape memory alloy (SMA) has been widely used in different applications due to its unique shape memory property. However, when used as an actuator, it exhibits a hysteresis behavior in its relation between temperature and strain, which is highly nonlinear and difficult to control. Although studies have been conducted on establishing various constitutive models of SMA, it is still difficult to achieve the precise control of the SMA wire with the existing models. In this work, a new promising approach regarding the SMA control task as a reinforcement learning (RL) problem is proposed to address this issue, which does not require accurate mathematical models. Both RL and an improved method named deep reinforcement learning (DRL) are used to solve the problem of precise control of a 1-D SMA wire actuator, respectively. The simulation results indicate that with the DRL method, the agent can precisely control the output deformation of the SMA wire after only ten episodes of training. Compared with the DRL method, the RL agent can also achieve the same training target but with hundreds of training.