Machine learning method for computation of optimal transitions in magnetic nanosystems
https://doi.org/10.17586/2220-8054-2020-11-6-642-650
Abstract
Minimum energy path (MEP) is an important tool for computation of activation barriers and transition rates for magnetic systems. Recently, new methods for numeric computation of MEP were proposed based on conjugate gradient and L-BFGS methods [1] significantly improved convergence rate compared to nudged elastic band (NEB) method. Due to lack of strict mathematical theory for MEP optimization other more effective methods are expected to exist. In this article, we propose a machine learning based approach to search for MEP computation methods. We reformulate the NEB method as a differentiable transformation in the space of all paths parametrized by a family of metaparameters. Using rate of convergence as the loss function, we train NEB optimizer to find optimal metaparameters. This meta learning technique can be the basis for deriving new optimization methods for computing MEP and other non-classical optimization problems.
Supporting agencies: The study of classical optimization methods for transition state computation in the sections 1 and 2 was funded by Government of the Russian Federation (Grant 08-08). The development of meta learning algorithm for nudged elastic band method in the section 3 was funded by Russian Science Foundation (Grant 19-42-06302).
About the Authors
K. R. Bushuev
ITMO University
Russian Federation
Russian Federation
Kronverkskiy, 49, Saint Petersburg, 197101
I. S. Lobanov
ITMO University; Saint Petersburg State University
Russian Federation
Russian Federation
Kronverkskiy, 49, Saint Petersburg, 197101
Saint Petersburg, 198504
References
1. Ivanov A.V., Dagbartsson D., Tranchida J., Uzdin V.M., Jnsson H. Efficient optimization method for finding minimum energy paths of magnetic transitions. Journal of Physics: Condensed Matter, 2020, 32(34).
2. Everschor-Sitte K., Masell J.,Reeve R.M., Klaui M. Perspective: Magnetic skyrmions - Overview of recent progress in an active research field. J. Appl. Phys., 2018, 124, P. 240901.
3. Mittal S. A survey of techniques for architecting processor components using domain-wall memory. ACM Journal on Emerging Technologies in Computing Systems (JETC), 2016, 13(2), P. 1–25.
4. Parkin S.S.P., Hayashi M., Thomas L. Magnetic domain-wall racetrack memory. Science, 2008, 320(5873), P. 190–194.
5. Fert A., Cros V., Sampaio J. Skyrmions on the track. Nature Nanotech, 2013, 8, P. 152–156.
6. Schratzberger J., Lee J., Fuger M., Fidler J., Fiedler G., Schref T., Suess D. Validation of the transition state theory with Langevin-dynamics simulations. Journal of Applied Physics, 2010, 108, P. 033915.
7. Desplat L., Suess D., Kim J-V., Stamps R.L. Thermal stability of metastable magnetic skyrmions: Entropic narrowing and significance of internal eigenmodes. Phys. Rev. B, 2018, 98, P. 134407.
8. Bessarab P.F., Mller G.P., Lobanov I.S. et. al. Lifetime of racetrack skyrmions. Sci. Rep., 2018, 8, P. 3433.
9. Potkina M.N., Lobanov I.S., Jnsson H., Uzdin V.M. Skyrmions in antiferromagnets: Thermal stability and the effect of external field and impurities. Journal of Applied Physics, 2020, 127, P. 213906.
10. Uzdin V.M., Potkina M.N., Lobanov I.S., Bessarab P.F., Jnsson H. The effect of confinement and defects on the thermal stability of skyrmions. Physica B: Condensed Matter, 2018, 549, P. 6–9.
11. Lobanov I.S., Jnsson H., Uzdin V.M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B., 2016, 94, P. 174418-2016.
12. Bessarab P.F., Uzdin V.M., Jnsson H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comp. Phys. Commun., 2015, 196, P. 335–347.
13. Bessarab P.F., Uzdin V.M., and Jnsson H. Harmonic transition-state theory of thermal spin transitions. Phys. Rev. B, 2012, 85, P. 184409.
14. Langer J.S. Statistical theory of the decay of metastable states. Annals of Physics, 1969, 54(2), P. 258–275.
15. Liashko S.Y., Lobanov I.S., Uzdin V.M., Jnsson H. Thermal stability of magnetic states in submicron magnetic islands. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8(5), P. 572–578.
16. Lobanov I.S., Uzdin V.M. The lifetime of big size topological chiral magnetic states. Estimation of the pre-exponential factor in the Arrhenius law. 2020. arXiv:2008.06754
17. Henkelman G., Jnsson H. A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J. Chem. Phys., 1999, 111(15), P. 7010–7022.
18. Olsen R.A., Kroes G.J., Henkelman G., Arnaldsson A., Jnsson H. Comparison of methods for finding saddle points without knowledge of the final states. J. Chem. Phys., 2004, 121(20), P. 9776–9792.
19. Gutierrez M.P., Argaez C., Jnsson H. Improved minimum mode following method for finding first order saddle points. J. Chem. Theo. Comput., 2017, 13(1), P. 125–134.
20. Henkelman G., Johannesson G., Jnsson H. Methods for finding saddle points and minimum energy paths. Theoretical Methods in Condensed Phase Chemistry, 2002, 5, P. 269–302.
21. Henkelman G., Uberuaga B.P., Jnsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys., 2000, 113(22), P. 9901–9904.
22. Maragakis P., Andreev S.A., Brumer Y., Reichman D.R., Kaxiras E. Adaptive nudged elastic band approach for transition state calculation. J. Chem. Phys., 2002, 117, P. 4651–4658.
23. Hoffmann M.,Mller G.P., Blgel S. Atomistic Perspective of Long Lifetimes of Small Skyrmions at Room Temperature. Phys. Rev. Lett., 2020, 124, P. 247201.
24. Henkelman G., Jnsson H. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J. Chem. Phys., 2000, 113(22), P. 9978–9985.
25. Weinan E., Weiqing R., Vanden-Eijnden E. String method for the study of rare events. Phys. Rev. B., 2002, 66, P. 052301(4).
26. Sheppard D., Terrell R., Henkelman G. Optimization methods for finding minimum energy paths. J. Chem. Phys., 2008, 128, P. 134106(10).
27. Heistracher P., Abert C., Bruckner F., Vogler C., Suess D., GPU-Accelerated Atomistic Energy Barrier Calculations of Skyrmion Annihilations. IEEE Transactions on Magnetics, 2018, 54(11), P. 1-5.
28. Schmidhuber J. Evolutionary Principles in Self-Referential Learning. On Learning how to Learn: The Meta-Meta-Meta...-Hook. PhD thesis, Institut f. Informatik, Tech. Univ. Munich, 1987.
29. Naik D.K., Mammone R.J. Meta-neural networks that learn by learning. In International Joint Conference on Neural Networks, IEEE, 1992, 1, P. 437–442.
30. Bengio S., Bengio Y., Cloutier J. On the search for new learning rules for ANNs. Neural Processing Letters, 1995, 2(4), P. 26–30.
31. Schmidhuber J., Zhao J., Wiering M. Shifting Inductive Bias with Success-Story Algorithm, Adaptive Levin Search, and Incremental SelfImprovement. Machine Learning, 1997, 28, P. 105–130.
32. Hochreiter S., Schmidhuber J. Long Short-Term Memory. Neural Computation, 1997.
33. Thrun S., Pratt L. Learning to learn. Springer Science and Business Media, 1998.
34. Younger A.S., Conwell P.R., Cotter N.E. Fixed-weight on-line learning. Transactions on Neural Networks, 1999, 10(2), P. 272–283.
35. Runarsson T.P., Jonsson M.T. Evolution and design of distributed learning rules. In IEEE Symposium on Combinations of Evolutionary Computation and Neural Networks, 2000, P. 59–63.
36. Diederik P. Kingma, Jimmy Ba. Adam: A Method for Stochastic Optimization. 3rd International Conference for Learning Representations, San Diego, 2015.
37. Andrychowicz M., Denil M., Gomez S., Hoffman M.W., Pfau D., Schaul T., Shillingford B., de Freitas N. Learning to learn by gradient descent by gradient descent. 2016, arXiv:1606.04474.
38. Chen Y., Hoffman M.W., Gomez S.C., Denil M., Lillicrap T.P., Botvinick M., de Freitas N. Learning to Learn without Gradient Descent by Gradient Descent. 2016, arXiv:1611.03824.
39. Wichrowska O., Maheswaranathan N., Hoffman M.W., Gomez S.C., Denil M., de Freitas N., Sohl-Dickstein J. Learned Optimizers that Scale and Generalize. 2017, arXiv:1703.04813.
40. Hospedales T., Antoniou A., Micaelli P., Storkey A. Meta-Learning in Neural Networks: A Survey. 2020, arXiv:2004.05439.
41. Carlon A., Espath L., Lopez R., Tempone R. Multi-iteration stochastic optimizers. 2020, arXiv:2011.01718
42. Khromova K. Optimization of the neural network training method. Master dissertation. ITMO University, 2020.
43. Rybakov F.N., Borisov A.B., Blgel S., Kiselev N.S. New Type of Stable Particlelike States in Chiral Magnets. Phys. Rev. Lett., 2015, 115, P. 117201.
44. Mentink J.H., Tretyakov M.V., Fasolino A., Katsnelson M.I., Rasing Th. Stable and fast semi-implicit integration of the stochastic LandauLifshitz equation. Journal of Physics: Condensed Matter, 2010, 22(17), P. 176001.
45. Desplat L., Vogler C., Kim J.-V., Stamps R. L., Suess D. Path sampling for lifetimes of metastable magnetic skyrmions and direct comparison with Kramers’ method. Phys. Rev. B., 2020, 101, P. 060403(R).
46. Moskalenko M.A., Lobanov I.S., Uzdin V.M. Demagnetizing fields in chiral magnetic structures. Nanosystems: Physics, Chemistry, Mathematics, 2020, 11(4), P. 401–407.
47. Jonsson H., Mills G., Jacobsen K.W., Berne B.J., Ciccotti G., Coker D.F. Nudged elastic band method for finding minimum energy paths of transitions, in Classical and Quantum Dynamics in Condensed Phase Simulations. World Scientific, Singapore, 1998, P. 385–404.
48. Henkelman G., Arnaldsson A., Jonsson H. Theoretical calculations of CH4 and H2 associative desorption from Ni(111): Could subsurface hydrogen play an important role? J. Chem. Phys., 2006, 124, P. 044706(9).
49. Einarsdottir D.M., Arnaldsson A., Oskarsson F., Jonsson H. Path optimization with application to tunneling. Lecture Notes in Computer Science, 2012, 7134, P. 45–55.
50. Malottki S.V., Dupe B., Bessarab P.F., Delin A., Heinze S. Enhanced skyrmion stability due to exchange frustration. Sci. Rep., 2017, 7(10), P. 12299.
51. Lobanov I.S., Potkina M.N., Jnsson H., Uzdin V.M. Truncated minimum energy path method for finding first order saddle points. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8(5), P. 586–595
52. Bradbury J., Frostig R., Hawkins P., Johnson M.J., Leary C., Maclaurin D., Necula G., Paszke A., Vanderlas J., Wanderman-Milne S., Zhang Q. JAX: composable transformations of Python+NumPy programs. (http://github.com/google/jax), 2018.
53. Mills G., Jonsson H., Schenter G.K. Reversible work based transition state theory: Application to H2 dissociative adsorption. Surf. Sci., 1995, 324, P. 305–337.
54. Zhu T., Li J., Samanta A., Kim H.G., Suresh S. Interfacial plasticity governs strain rate sensitivity and ductility in nanostructured metals. PNAS, 2007, 104(9), P. 3031–3036.
55. Fakoor M., Kosari A., Jafarzadeh M. Revision on fuzzy artificial potential field for humanoid robot path planning in unknown environment. International Journal of Advanced Mechatronic Systems (IJAMECHS), 2015, 6(4), P. 174–183
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Bushuev K.R., Lobanov I.S. Machine learning method for computation of optimal transitions in magnetic nanosystems. Nanosystems: Physics, Chemistry, Mathematics. 2020;11(6):642–650. https://doi.org/10.17586/2220-8054-2020-11-6-642-650
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