This paper presents a learning-based framework for approximating an exact magnetic-field interaction model, supported by both numerical and experimental validation. High-fidelity magnetic-field interaction modeling is essential for achieving exceptional accuracy and responsiveness across a wide range of fields, including transportation, energy systems, medicine, biomedical robotics, and aerospace robotics. In aerospace engineering, magnetic actuation has been investigated as a fuel-free solution for multi-satellite attitude and formation control. Although the exact magnetic field can be computed from the Biot-Savart law, the associated computational cost is prohibitive, and prior studies have therefore relied on dipole approximations to improve efficiency. However, these approximations lose accuracy during proximity operations, leading to unstable behavior and even collisions. To address this limitation, we develop a learning-based approximation framework that faithfully reproduces the exact field while dramatically reducing computational cost. This framework directly derives a coefficient matrix that maps inter-satellite current vectors to the resulting forces and torques, enabling efficient computation of control current commands. The proposed method additionally provides a certified error bound, derived from the number of training samples, ensuring reliable prediction accuracy. The learned model can also accommodate interactions between coils of different sizes through appropriate geometric transformations, without retraining. To verify the effectiveness of the proposed framework under challenging conditions, a spacecraft docking scenario is examined through both numerical simulations and experimental validation.

翻译：本文提出了一种基于学习的框架，用于逼近精确的磁场交互模型，并通过数值计算与实验验证提供支持。高精度磁场交互建模对于交通、能源系统、医学、生物医学机器人及航天机器人等多个领域实现卓越精度与响应能力至关重要。在航天工程中，磁驱动已被研究作为多卫星姿态与编队控制的无燃料解决方案。尽管精确磁场可通过Biot-Savart定律计算得出，但其计算成本过高，因此先前研究多依赖偶极子近似以提高效率。然而，此类近似在近距操作时会丧失精度，导致不稳定行为甚至碰撞。为解决这一局限，我们开发了一种基于学习的逼近框架，既能忠实还原精确磁场，又能大幅降低计算成本。该框架直接导出耦合系数矩阵，将星间电流向量映射为作用力与力矩，从而高效计算控制电流指令。所提方法还基于训练样本数量提供了认证型误差界，确保预测精度的可靠性。学习模型还可通过适当的几何变换适应不同尺寸线圈间的交互，无需重新训练。为验证所提框架在挑战性条件下的有效性，我们通过数值仿真与实验验证对航天器对接场景进行了研究。