Network function virtualization (NFV) implements network functions as Virtual Network Functions (VNFs) to address the challenges faced by traditional network infrastructures. Service Function Chaining (SFC) chains VNFs in a specific order to process packets, but placing SFCs across multiple domains while ensuring reliability is challenging. Existing methods are time-consuming and error-prone. In this paper, we study the reliability aware multi-domain SFC placement problem. The reliability aware multi-domain SFC placement problem is formulated as a multi-objective optimization model to minimize SFC placement resource consumption cost (SPRC) and SFC Placement operating cost (SPOC) while satisfying the specific reliability requirements. Then, we propose a Deep Reinforcement Learning-based SFC placement algorithm, called DRL-SFCP, to optimize the reliability of SFC placement in multi-domain networks. Finally, we conduct extensive experiments and results demonstrate that our proposed placement method outperforms existing approaches in terms of SPRC and SPOC.