Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations
复制标题
DOI：
10.1007/s40304-017-0117-6
复制
发表时间：
2017-12-01
影响因子：
0.9
通讯作者：
Jentzen, Arnulf
中科院分区：
文献类型：
--
作者：
E, Weinan;Han, Jiequn;Jentzen, Arnulf
我们研究了一种用于求解抛物线偏微分方程（PDE）和向后随机微分方程（BSDE）的新算法，高维度是基于BSDE和增强学习之间的类比，其解决方案的作用扮演了策略的作用。功能，以及由规定的终端条件和BSDE解决方案之间的误差给出的损耗函数。然后，像深度强化学习一样，神经网络近似策略功能。使用TensorFlow的数值结果说明了研究算法的效率和准确性，用于从物理和金融等几种100维非线性PDE（例如Allen-Cahn方程），汉密尔顿 - 贾科比·贝尔曼方程，以及金融衍生品的非线性定价模型。
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.