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Solving for high-dimensional committor functions using artificial neural networks

  • Yuehaw KhooEmail author
  • Jianfeng Lu
  • Lexing Ying
Research
  • 34 Downloads

Abstract

In this note we propose a method based on artificial neural network to study the transition between states governed by stochastic processes. In particular, we aim for numerical schemes for the committor function, the central object of transition path theory, which satisfies a high-dimensional Fokker–Planck equation. By working with the variational formulation of such partial differential equation and parameterizing the committor function in terms of a neural network, approximations can be obtained via optimizing the neural network weights using stochastic algorithms. The numerical examples show that moderate accuracy can be achieved for high-dimensional problems.

References

  1. 1.
    Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, J., Levenberg, M., Mane, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viegas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: Tensorflow: large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467 (2016)
  2. 2.
    Berg, J., Nyström, K.: A unified deep artificial neural network approach to partial differential equations in complex geometries. arXiv preprint arXiv:1711.06464 (2017)
  3. 3.
    Carleo, G., Troyer, M.: Solving the quantum many-body problem with artificial neural networks. Science 355(6325), 602–606 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Coifman, R.R., Kevrekidis, I.G., Lafon, S., Maggioni, M., Nadler, B.: Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems. Multiscale Model. Simul. 7(2), 842–864 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kingma, D., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
  8. 8.
    Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987–1000 (1998)CrossRefGoogle Scholar
  9. 9.
    Lai, R., Lu, J.: Point cloud discretization of Fokker-Planck operators for committor functions. Multiscale Model. Simul. arXiv preprint arXiv:1703.09359 (2017) (in press)
  10. 10.
    LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015)CrossRefGoogle Scholar
  11. 11.
    Lu, J., Nolen, J.: Reactive trajectories and the transition path process. Probab. Theory Relat. Fields 161, 195–244 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lu, J., Nolen, J.: Reactive trajectories and the transition path process. Probab. Theory Relat. Fields 161(1–2), 195–244 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)CrossRefGoogle Scholar
  14. 14.
    Sirignano, J., Spiliopoulos, K.: DGM: a deep learning algorithm for solving partial differential equations. arXiv preprint arXiv:1708.07469 (2017)
  15. 15.
    Vanden-Eijnden, E., Venturoli, M.: Revisiting the finite temperature string method for the calculation of reaction tubes and free energies. J. Chem. Phys. 130, 194103 (2009)CrossRefGoogle Scholar
  16. 16.
    E, W., Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5(4), 349–380 (2017)Google Scholar
  17. 17.
    E, W., Ren, W., Vanden-Eijnden, E.: Finite temparture string method for the study of rare events. J. Phys. Chem. B 109, 6688–6693 (2005)Google Scholar
  18. 18.
    E, W., Vanden-Eijnden, E.: Towards a theory of transition paths. J. Stat. Phys. 123(3), 503 (2006)Google Scholar
  19. 19.
    E, W., Vanden-Eijnden, E.: Transition path theory and path-finding algorithms for the study of rare events. Ann. Rev. Phys. Chem. 61, 391–420 (2010)Google Scholar
  20. 20.
    E, W., Yu, B.: The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1–12 (2018)Google Scholar
  21. 21.
    Zhang, L., Wang, H., E, W.: Reinforced dynamics of large atomic and molecular systems. arXiv preprint arXiv:1712.03461 (2017)

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of Mathematics, Department of Chemistry and Department of PhysicsDuke UniversityDurhamUSA
  3. 3.Department of Mathematics and ICMEStanford UniversityStanfordUSA

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