The simulation of magnetic materials has played a critical role in the prediction of magnetization dynamics and novel magnetization states. Simulations could be further enhanced by utilizing machine learning techniques. However, their implementation depend on the task at hand, and it is especially desirable for the learning to be explainable [1]. One way to achieve this goal is by using models that are better suited to interpret the results from machine learning. From this perspective, magnetism presents the fundamental challenge of atomic- and micrometer-range phenomena. To work towards solving this issue, we have proposed a variation of the micromagnetic model called the pseudospectral Landau-Lifshitz equation, PS-LL [2]. This approach interprets the exchange interaction in Fourier space, specifically as a convolution kernel that can be either provided [3] or derived from first principles [4]. The model is by definition grid-independent, smoothly transitioning from atomic interactions to the continuum approximation. With this model, the learning occurs directly in the dispersion relation so that predictions can be interpreted. As an example, we discuss how to apply the PS-LL model to train a convolutional neural network (CNN) with the ability to predict defect’s mean size and density for a particular outcome, in this case, an average domain-wall width [5]. This technique could be further extended to higher dimensions and combined with other types of neural networks to achieve the desired predictions. More importantly, the use of the PS-LL model as an underlying framework makes it possible to develop explainable networks and predictions returning physical quantities and their errors, which are vital for any experimental realization and feasibility analysis of new materials.

References

[1] X. Zhong et al., npj Computational Materials 8, 204 (2022)
[2] K. Rockwell et al., Phys. Rev. B 108, L180404 (2024)
[3] A. Roxburgh, M. Copus, and E. Iacocca, arXiv:2506:19572
[4] M. Copus and E. Iacocca, arXiv: 2506:14365
[5] C. Eagan, M. Copus, and E. Iacocca, in preparation.