Abstract
Understanding how complex systems transition between states requires mapping the energy landscape that governs these changes. Local transition-state networks reveal the barrier architecture that explains observed behaviour and enables mechanism-based prediction across computational chemistry, biology, and physics, yet in many practical settings current approaches either require pre-specified endpoints or rely on single-ended searches that provide only a limited sample of nearby saddles. We present a general optimization framework that systematically expands local coverage by coupling a multi-objective explorer with a bilayer minimum-mode kernel. The inner layer uses Hessian-vector products to recover the lowest-curvature subspace, the outer layer optimizes on a reflected force to reach index-1 saddles, then a two-sided descent certifies connectivity. The GPU-based pipeline is portable across autodiff backends and eigensolvers and, on large atomistic-spin tests, matches explicit-Hessian accuracy while cutting peak memory and wall time by orders of magnitude. Applied to a DFT-parameterized Néel-type skyrmionic model, it recovers known routes and reveals previously unreported mechanisms, including meron-antimeron-mediated Néel-type skyrmionic duplication, annihilation, and chiral-droplet formation, enabling up to 32 pathways between biskyrmion (Q = 2) and biantiskyrmion (Q = −2). The same core transfers to Cartesian atoms, automatically mapping canonical rearrangements of a Ni(111) heptamer, underscoring the framework’s generality.
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Data availability
All data supporting the findings of this study are available within the paper and its Supplementary Information.Code availability:The reported results were obtained using standard open-source libraries. The optimization scheme described in the Methods and Supplementary Information was implemented with automatic differentiation operators in PyTorch, the NSGA-II algorithm as provided in the pyMOO package, atomistic spin dynamics simulations with UppASD, and structural relaxations with the Atomic Simulation Environment (ASE). The full algorithmic workflow is explicitly provided in Supplementary Note 1, which allows independent re-implementation using these widely available packages. A complexity and scalability discussion is provided in Supplementary Note 3.
Code availability
The reported results were obtained using standard open-source libraries. The optimization scheme described in the Methods and Supplementary Information was implemented with automatic differentiation operators in PyTorch, the NSGA-II algorithm as provided in the pyMOO package30,31, atomistic spin dynamics simulations with UppASD38, and structural relaxations with the Atomic Simulation Environment (ASE)39. The full algorithmic workflow is explicitly provided in Supplementary Note 1, which allows independent re-implementation using these widely available packages. A complexity and scalability discussion is provided in Supplementary Note 3.
References
Wales, D. J. Energy landscapes: Calculating pathways and rates. J. Phys. Chem. B 110, 20765–20776 (2006).
Wales, D. J. Exploring energy landscapes. Annu. Rev. Phys. Chem. 69, 401–425 (2018).
Binder, K. & Young, A. P. Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986).
Onuchic, J. N., Luthey-Schulten, Z. & Wolynes, P. G. Theory of protein folding: The energy landscape perspective. Annu. Rev. Phys. Chem. 48, 545–600 (1997).
Bussi, G., Laio, A. & Parrinello, M. Using metadynamics to explore complex free-energy landscapes. Nat. Rev. Phys. 2, 200–212 (2020).
Shires, B. W. B. & Pickard, C. J. Visualizing energy landscapes through manifold learning. Phys. Rev. X 11, 041026 (2021).
Henkelman, G. & Jónsson, H. A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys. 111, 7010–7022 (1999).
Henkelman, G. & Jónsson, H. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J. Chem. Phys. 113, 9978–9985 (2000).
Bessarab, P. F., Uzdin, V. M. & Jónsson, H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Computer Phys. Commun. 196, 335–347 (2015).
Becker, O. M. & Karplus, M. The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics. J. Chem. Phys. 106, 1495–1517 (1997).
Ceriotti, M., Tribello, G. A. & Parrinello, M. Simplifying the representation of complex free-energy landscapes using sketch-map. Proc. Natl. Acad. Sci. 108, 13023–13028 (2011).
Wales, D. J. Dynamical signatures of multifunnel energy landscapes. J. Phys. Chem. Lett. 13, 6349–6358 (2022).
Sheppard, D., Terrell, R. & Henkelman, G. Optimization methods for finding minimum energy paths. J. Chem. Phys. 128, 134106 (2008).
Trygubenko, S. A. & Wales, D. J. A doubly nudged elastic band method for finding transition states. J. Chem. Phys. 120, 2082–2094 (2004).
Munro, L. J. & Wales, D. J. Defect migration in crystalline silicon. Phys. Rev. B 59, 3969–3980 (1999).
Heyden, A., Bell, A. T. & Keil, F. J. Efficient methods for finding transition states in chemical reactions: Comparison of improved dimer method and partitioned rational function optimization method. J. Chem. Phys. 123, 224101 (2005).
Zeng, Y., Xiao, P. & Henkelman, G. Unification of algorithms for minimum mode optimization. J. Chem. Phys. 140, 044115 (2014).
Wales, D. J. PATHSAMPLE: A driver for OPTIM to create stationary point databases using discrete path sampling and perform kinetic analysis. https://www-wales.ch.cam.ac.uk/PATHSAMPLE/. Accessed 2025-12-31.
Wales, D. J. Discrete path sampling. Mol. Phys. 100, 3285–3305 (2002).
Wales, D. J. Some further applications of discrete path sampling to cluster isomerization. Mol. Phys. 102, 891–908 (2004).
Carr, J. M. & Wales, D. J. Global optimization and folding pathways of selected α-helical proteins. J. Chem. Phys. 123, 234901 (2005).
Schrautzer, H., Sallermann, M., Bessarab, P. F. & Jónsson, H. Identification of mechanisms of magnetic transitions using an efficient method for converging on first-order saddle points. Phys. Rev. B 112, 104433 (2025).
Miranda, I. P. et al. Band-filling effects on the emergence of magnetic skyrmions: Pd/Fe and Pd/Co bilayers on Ir(111). Phys. Rev. B 105, 224413 (2022).
Xu, Q. et al. Metaheuristic conditional neural network for harvesting skyrmionic metastable states. Phys. Rev. Res. 5, 043199 (2023).
Xu, Q., Shen, Z. & Delin, A. Design of 2d skyrmionic metamaterials through controlled assembly. npj Computational Mater. 11, 56 (2025).
Xu, Q. et al. Genetic-tunneling driven energy optimizer for spin systems. Commun. Phys. 6, 239 (2023).
Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions. Nat. Phys. 7, 713–718 (2011).
Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639 (2013).
Deb, K.Multi-Objective Optimization Using Evolutionary Algorithms (John Wiley & Sons, 2001).
Deb, K., Pratap, A., Agarwal, S. & Meyarivan, T. A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE Trans. Evolut. Comput. 6, 182–197 (2002).
Blank, J. & Deb, K. pymoo: Multi-objective optimization in python. IEEE Access 8, 89497–89509 (2020).
Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45, 255–282 (1950).
Knyazev, A. V. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001).
Foiles, S., Baskes, M. & Daw, M. S. Embedded-atom-method functions for the fcc metals cu, ag, au, ni, pd, pt, and their alloys. Phys. Rev. B 33, 7983 (1986).
Carr, J. M., Trygubenko, S. A. & Wales, D. J. Finding pathways between distant local minima. J. Chem. Phys. 122, 234903 (2005).
Wales, D. J. & Carr, J. M. A quasi-continuous interpolation scheme for pathways between distant configurations. J. Chem. Theory Comput. 8, 5020–5034 (2012).
Griffiths, M., Niblett, S. P. & Wales, D. J. Optimal alignment of structures for finite and periodic systems. J. Chem. Theory Comput. 13, 4914–4931 (2017).
Skubic, B., Hellsvik, J., Nordström, L. & Eriksson, O. A method for atomistic spin dynamics simulations: implementation and examples. J. Phys.: Condens. Matter 20, 315203 (2008).
Larsen, A. H. et al. The atomic simulation environment – a python library for working with atoms. J. Phys.: Condens. Matter 29, 273002 (2017).
Holt, C. C. Forecasting seasonals and trends by exponentially weighted moving averages. Int. J. Forecast. 20, 5–10 (2004).
Acknowledgements
The authors thank Filipp N. Rybakov (Uppsala University), Pavel Bessarab (Linnaeus University) and Mathias Augustin (Uppsala University) for many fruitful discussions. We also thank Johan Hellsvik (KTH, PDC Center for High Performance Computing) for his support with GPU resources. The authors used AI-assisted tools to improve the language of the manuscript.Financial support from theSwedish Research Council (Vetenkapsrådet, VR) Grant No. 2016-05980, Grant No. 2019-05304, and Grant No. 2024-04986, and the Knut and Alice Wallenberg foundation Grant No. 2018.0060, Grant No. 2021.0246, and Grant No. 2022.0108 is acknowledged. The Wallenberg Initiative Materials Science for Sustainability (WISE) funded by the Knut and Alice Wallenberg Foundation is also acknowledged.The computations/data handling were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the Swedish Research Council through grant agreement no. 2022-06725.
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Q.C. conceived the idea, carried out the research, A.D. supervised the project. Both authors contributed to the interpretation of the results and to the writing and revision of the manuscript.
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Xu, Q., Delin, A. A general optimization framework for mapping local transition-state networks. npj Comput Mater (2026). https://doi.org/10.1038/s41524-026-01985-3
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DOI: https://doi.org/10.1038/s41524-026-01985-3
