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Solving for high-dimensional committor functions using artificial neural networks

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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.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Yuehaw Khoo

  2. Department of Mathematics, Department of Chemistry and Department of Physics, Duke University, Durham, NC, 27708, USA

    Jianfeng Lu

  3. Department of Mathematics and ICME, Stanford University, Stanford, CA, 94305, USA

    Lexing Ying

Authors
  1. Yuehaw Khoo
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  2. Jianfeng Lu
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  3. Lexing Ying
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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

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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

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