Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochas
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Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations Weinan E1,2,3 · Jiequn Han2 · Arnulf Jentzen4
Received: 19 July 2017 / Accepted: 23 August 2017 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany 2017
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. Keywords PDEs · High dimension · Backward stochastic differential equations · Deep learning · Control · Feynman-Kac Mathematics Subject Classification 65M75 · 60H35 · 65C30
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Weinan E [email protected] Jiequn Han [email protected] Arnulf Jentzen [email protected]
1
Beijing Institute of Big Data Research, Beijing, China
2
Princeton University, Princeton, NJ, USA
3
Peking University, Beijing, China
4
ETH Zurich, Zurich, Switzerland
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W. E et al.
1 Introduction Developing efficient numerical algorithms for high-dimensional (say, hundreds of dimensions) partial differential equations (PDEs) has been one of the most challenging tasks in applied mathematics. As is well known, the difficulty lies in the “curse of dimensionality” [1], namely, as the dimensionality grows, the complexity of the algorithms grows exponentially. For this reason, there are only a limited number of cases where practical high-dimensional algorithms have been developed (cf., e.g., [9,12,13,20–22]). For linear parabolic PDEs, one can use the Feynman–Kac formula and Monte Carlo methods to develop efficient algorithms to evaluate solutions at any given space–time locations. For a class of inviscid Hamilton–Jacobi equations, Darbon and Osher have recently developed an algorithm which performs numerically well in the case of such high-dimensional inviscid Hamilton–Jacobi equations (see [9]). Darbon and Osher’s algorithm is based on results from compressed sensing and on the Hopf formulas for the Hamilton–Jacobi equations. A general algorithm for (nonlinear) parabolic PDEs based on the Feynman–Kac and Bismut–Elworthy–Li formula and a multilevel decomposition of Picard iteration was developed in [12] and has been shown to be quite efficient on a number examples in finance and physics (cf. [13]). The complexity of the algorithm is essentially s
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