In this paper, we propose a deep learning framework for solving high-dimensional partial integro-differential equations (PIDEs) based on the temporal difference learning. We introduce a set of Levy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and non-local terms of the equations. Subsequently, we train the networks using the temporal difference error, termination condition, and properties of the non-local terms as the loss function. The relative error of the method reaches O(10^{-3}) in 100-dimensional experiments and O(10^{-4}) in one-dimensional pure jump problems. Additionally, our method demonstrates the advantages of low computational cost and robustness, making it well-suited for addressing problems with different forms and intensities of jumps.

翻译：本文提出一种基于时域差分学习的深度学习框架，用于求解高维偏积分微分方程。我们引入一组Levy过程并构建相应的强化学习模型。通过深度神经网络表示方程的解与非局部项，模拟整个计算过程。随后利用时域差分误差、终止条件以及非局部项的性质构建损失函数对网络进行训练。该方法在100维实验中的相对误差达到O(10^{-3})，在一维纯跳跃问题中达到O(10^{-4})。此外，该方法具有计算成本低和鲁棒性强的优势，特别适用于处理不同形式和强度的跳跃问题。