论文标题
用于高维偏微分方程的Actor-Critic算法
Actor-Critic Algorithm for High-dimensional Partial Differential Equations
论文作者
论文摘要
我们开发了一个深度学习模型，以有效解决高维非线性抛物线偏微分方程（PDE）。我们遵循Feynman-kac公式，将PDE重新调整为受后向随机微分方程（BSDE）系统控制的等效随机控制问题。 BSDE的Markovian属性用于设计我们的神经网络体系结构，该建筑的灵感来自参与者批评算法，通常用于深入增强学习。与最先进的模型相比，我们进行了几项改进，包括1）大大降低了可训练的参数，2）更快的收敛速度和3）调节的超参数较少。我们通过求解一些众所周知的PDE类别，例如汉密尔顿 - 雅各布 - 贝尔曼方程，艾伦 - 卡恩方程和黑色 - choles方程，以100的顺序来证明这些改进。
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.
