Actor-Critic Algorithm for High-dimensional Partial Differential Equations
This addresses computational challenges in solving high-dimensional PDEs for applications like finance and physics, representing an incremental improvement over existing methods.
The paper tackles solving high-dimensional nonlinear parabolic partial differential equations (PDEs) by reformulating them into stochastic control problems using the Feynman-Kac formula and designing a neural network based on the Actor-Critic algorithm, achieving improvements including reduced trainable parameters, faster convergence, and fewer hyperparameters for PDEs with dimensions up to 100.
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.