Actor-Critic Algorithm for High-dimensional Partial Differential Equations

• URL: http://arxiv.org/abs/2010.03647v1
• Date: Wed, 7 Oct 2020 20:53:24 GMT
• Title: Actor-Critic Algorithm for High-dimensional Partial Differential Equations
• Authors: Xiaohan Zhang
• Abstract summary: We develop a deep learning model to solve high-dimensional nonlinear parabolic partial differential equations. The Markovian property of the BSDE is utilized in designing our neural network architecture. We demonstrate those improvements by solving a few well-known classes of PDEs.
• Score: 1.5644600570264835
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a deep learning model to effectively solve high-dimensional nonlinear parabolic partial differential equations (PDE). We follow Feynman-Kac formula to reformulate PDE into the equivalent stochastic control problem governed by a Backward Stochastic Differential Equation (BSDE) system. The Markovian property of the BSDE is utilized in designing our neural network architecture, which is inspired by the Actor-Critic algorithm usually applied for deep Reinforcement Learning. Compared to the State-of-the-Art model, we make several improvements including 1) largely reduced trainable parameters, 2) faster convergence rate and 3) fewer hyperparameters to tune. We demonstrate those improvements by solving a few well-known classes of PDEs such as Hamilton-Jacobian-Bellman equation, Allen-Cahn equation and Black-Scholes equation with dimensions on the order of 100.

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