Cooperative multi-agent reinforcement learning (MARL) aims to coordinate a system of agents towards optimizing global returns (Vinyals et al., 2019), and has witnessed significant prospects in various domains, such as autonomous vehicles (Zhou et al., 2020), smart grid (Chen et al., 2021a), robotics (Yu et al., 2023), and social science (Leibo et al., 2017). Training reliable control policies for coordinating such systems remains a major challenge. The centralized training with decentralized execution (CTDE) (Foerster et al., 2016) hybrids the merits of independent Q-learning (Foerster et al., 2017) and joint action learning (Sukhbaatar et al., 2016), and becomes a compelling paradigm that exploits the centralized training opportunity for training fully decentralized policies (Wang et al., 2023). Subsequently, numerous popular algorithms are proposed, including VDN (Sunehag et al., 2018), QMIX (Rashid et al., 2020), MAAC (Iqbal & Sha, 2019), and MAPPO (Yu et al., 2022). Sharing policy parameters is crucial for scaling these algorithms to massive agents with accelerated cooperation learning (Fu et al., 2022). However, it is widely observed that agents tend to acquire homogeneous behaviors, which might hinder diversified exploration and sophisticated coordination (Christianos et al., 2021). Some methods (Li et al., 2021; Jiang & Lu, 2021; Liu et al., 2023) attempt to promote individualized behaviors by distinguishing each agent from the others, while they often neglect the prospect of effective team composition with implicit task allocation. Real-world multi-agent tasks usually involve dynamic team composition with the emergence of roles (Shao et al., 2022; Hu et al., 2022).

Goal-conditioned reinforcement learning (RL) is an interesting extension of the traditional RL framework, where the dynamic environment and reward sparsity can cause conventional learning algorithms to fail. Reward shaping is a practical approach to improving sample efficiency by embedding human domain knowledge into the learning process. Existing reward shaping methods for goal-conditioned RL are typically built on distance metrics with a linear and isotropic distribution, which may fail to provide sufficient information about the ever-changing environment with high complexity. This paper proposes a novel magnetic field-based reward shaping (MFRS) method for goal-conditioned RL tasks with dynamic target and obstacles. Inspired by the physical properties of magnets, we consider the target and obstacles as permanent magnets and establish the reward function according to the intensity values of the magnetic field generated by these magnets. The nonlinear and anisotropic distribution of the magnetic field intensity can provide more accessible and conducive information about the optimization landscape, thus introducing a more sophisticated magnetic reward compared to the distance-based setting. Further, we transform our magnetic reward to the form of potential-based reward shaping by learning a secondary potential function concurrently to ensure the optimal policy invariance of our method. Experiments results in both simulated and real-world robotic manipulation tasks demonstrate that MFRS outperforms relevant existing methods and effectively improves the sample efficiency of RL algorithms in goal-conditioned tasks with various dynamics of the target and obstacles.