Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

Weinan E, Jiequn Han, Arnulf Jentzen

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on 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 studied algorithm 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.

Original languageEnglish (US)
Pages (from-to)349-380
Number of pages32
JournalCommunications in Mathematics and Statistics
Volume5
Issue number4
DOIs
StatePublished - Dec 1 2017

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Backward Stochastic Differential Equation
Parabolic Partial Differential Equations
Partial differential equations
Numerical methods
Differential equations
High-dimensional
Numerical Methods
Reinforcement learning
Reinforcement Learning
Financial Derivatives
Allen-Cahn Equation
Finance
Loss Function
Nonlinear Partial Differential Equations
Higher Dimensions
Pricing
Analogy
Physics
Neural Networks
Gradient

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics
  • Statistics and Probability

Cite this

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