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Workshop: The Symbiosis of Deep Learning and Differential Equations

Actor-Critic Algorithm for High-dimensional PDEs

Xiaohan Zhang

Abstract: We develop a machine learning model to effectively solve high-dimensional nonlinear parabolic partial differential equations (PDE). We use Feynman-Kac formula to reformulate PDE into the equivalent stochastic control problem governed by a Backward Stochastic Differential Equation (BSDE) system. Our model is designed to maximally exploit the Markovian property of the BSDE system and utilizes an Actor-Critic network architecture, which is novel in the high dimensional PDE literature. We show that our algorithm design leads to a significant speedup with higher accuracy level compared to other neural network solvers. Our model advances the state-of-the-art machine learning PDE solvers in a few aspects: 1) the trainable parameters are reduced by $N$ times, where $N$ is the number of steps to discretize the PDE in time, 2) the model convergence rate is an order of magnitude faster, 3) our model has fewer tuning hyperparameters. We demonstrate the performance improvements by solving six equations including Hamilton-Jacobian-Bellman equation, Allen-Cahn equation and Black-Scholes equation, all with dimensions on the order of 100. Those equations in high dimensions have wide applications in control theory, material science and Quantitative finance.

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