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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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