Solving for high-dimensional committor functions using artificial neural networks

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  11. 11.

    Lu, J., Nolen, J.: Reactive trajectories and the transition path process. Probab. Theory Relat. Fields 161, 195–244 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lu, J., Nolen, J.: Reactive trajectories and the transition path process. Probab. Theory Relat. Fields 161(1–2), 195–244 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)

    Article  Google 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)

    Article  Google 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)

  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)

  18. 18.

    E, W., Vanden-Eijnden, E.: Towards a theory of transition paths. J. Stat. Phys. 123(3), 503 (2006)

  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)

  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)

  21. 21.

    Zhang, L., Wang, H., E, W.: Reinforced dynamics of large atomic and molecular systems. arXiv preprint arXiv:1712.03461 (2017)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yuehaw Khoo.

Additional information

The work of Y.K. and L.Y. is supported in part by the National Science Foundation under Award DMS-1521830 and the U.S. Department of Energys Advanced Scientific Computing Research program under Award DE-FC02-13ER26134/DE-SC0009409. The work of J.L. is supported in part by the National Science Foundation under Award DMS-1454939. The collaboration is also supported by the National Science Foundation Research Networks in Mathematical Sciences KI-Net under Grants DMS-1107444 and DMS-1107465

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khoo, Y., Lu, J. & Ying, L. Solving for high-dimensional committor functions using artificial neural networks. Res Math Sci 6, 1 (2019). https://doi.org/10.1007/s40687-018-0160-2

Download citation