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. 2023 Nov 13;23(1):79–87. doi: 10.1038/s41563-023-01698-8

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© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 4 — a–c, Spin-wave spectra of the helical (a), skyrmion (b) and conical (c) magnetic phases. d, Sine-wave input sequence defining the applied field amplitudes. e–g, Spin-wave spectra of the helical (e), skyrmion (f) and conical (g) phases at nodes of the sine-wave input fields from d. h, H_c evolution of the MSE values at T = 4 K and H_range = 90 mT, for forecasting (MG(N + 10)) and transformation (square wave) target applications. Note that MSE' denotes the normalized scale of MSE for [0, 1], where 0 (1) represents the best (worst) MSE. A (meta)stable magnetic field range for each phase is colour-coded. i, MSE' and task-agnostic metric results as a function of H_c at T = 4 K. j, Correlation matrix of Spearman’s rank correlation coefficient. k, Performance of forecasting as an evolution of the MC metric. l, Performance of transformation as an evolution of the CP metric.