In this work, we explore advanced machine learning techniques for minimizing Gibbs free energy in full 3D micromagnetic simulations. Building on Brown's bounds for magnetostatic self-energy, we revisit their application in the context of variational formulations of the transmission problems for the scalar and vector potential. To overcome the computational challenges posed by whole-space integrals, we reformulate these bounds on a finite domain, making the method more efficient and scalable for numerical simulation. Our approach utilizes an alternating optimization scheme for joint minimization of Brown's energy bounds and the Gibbs free energy. The Cayley transform is employed to rigorously enforce the unit norm constraint, while R-functions are used to impose essential boundary conditions in the computation of magnetostatic fields. Our results highlight the potential of mesh-free Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs) when integrated with hard constraints, providing highly accurate approximations. These methods exhibit competitive performance compared to traditional numerical approaches, showing significant promise in computing magnetostatic fields and the application for energy minimization, such as the computation of hysteresis curves. This work opens the path for future directions of research on more complex geometries, such as grain structure models, and the application to large scale problem settings which are intractable with traditional numerical methods.

翻译：本文探索了在全三维微磁模拟中最小化吉布斯自由能的先进机器学习技术。基于布朗静磁自能界，我们重新审视了其在标量与矢量势传输问题变分表述中的应用。为克服全空间积分带来的计算挑战，我们在有限域上重构了这些界，使该方法在数值模拟中更具效率与可扩展性。我们的方法采用交替优化方案，联合最小化布朗能量界与吉布斯自由能。通过凯莱变换严格实施单位范数约束，同时利用R函数在静磁场计算中施加本质边界条件。研究结果表明，当与硬约束结合时，无网格物理信息神经网络（PINNs）与极限学习机（ELMs）能够提供高精度近似解。相较于传统数值方法，这些方法展现出具有竞争力的性能，在静磁场计算及能量最小化应用（如磁滞回线计算）中显示出巨大潜力。本工作为未来研究开辟了新方向，包括更复杂的几何结构（如晶粒结构模型）以及传统数值方法难以处理的大规模问题场景的应用。