Parallel processing approach to physics-informed machine learning for solving magnetostatic problems
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As machine learning methods based on physical laws, there are PINNs designed based on the universal approximation theorem for neural networks and KANs designed based on the Kolmogorov-Arnold representation theorem, both of which are widely studied. This research aims to apply such physics-informed machine learning methods to magnetostatic problems derived from Maxwell's equations, and to optimize the procedure for efficiency and high performance. The magnetostatic problem exhibits discontinuities at the boundaries between different materials, such as a magnetic core in air. In conventional numerical methods, it is known that the interface conditions of electromagnetic fields can be satisfied naturally by using Nedelec’s edge element in FEM and Yee’s grid in the FDTD method. However, PINN, which does not use connectivity between training points, cannot be expected to satisfy naturally. On the other hand, networks designed to represent arbitrary continuous functions struggle to express the solution function. Therefore, this research proposes a solution through network partitioning and further constructs a parallel algorithm.
