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Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

by W. E and J. Han and A. Jentzen

(Report number 2017-29)

Abstract
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several $100$-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

Keywords:

@Techreport{EHJ17_725,
  author = {W. E and J. Han and A. Jentzen},
  title = {Deep learning-based numerical methods for high-dimensional 
parabolic partial differential equations and backward stochastic 
differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-29},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-29.pdf },
  year = {2017}
}

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