Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman-Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

翻译：非线性部分差异方程式( PDEs) 用于在大量科学领域模拟动态进程, 从金融到生物学。 在许多应用中, 标准本地模型不足以准确解释某些非本地现象, 例如远程互动等。 为了正确捕捉这些现象, 文献中经常使用非本地非线性 PDE 模型。 在本篇文章中, 我们建议了两种基于机器学习和 Picard 迭代的数值方法, 以大致解决非本地非线性 PDE 。 提议的机器学习法是先前在文献中引入的基于深度学习的分解类型近似方法的扩展变异。 并且使用神经性能的神经性能网络为某些非本地现象提供近似的解决办法。 Picard 偏重法是所谓的完整历史的延伸变异种。 先前在文献中引入的 Piccar 近似方案, 为域的单一点提供了一种近似的解决办法。 两种方法都是无线性, 允许非本地的 PDE 偏近似方法, 先前在文献中引入了基于 Nemann 直径的不直径直径直径直径直径性近近近近近近近近近近近近近近近方法,,,, 的平面直径直径直径直径性直径直径直径直径直径直径直径直径,, 直径直径直到直到直到直到直到直径直径直径直到直到直至直到直到直到直到直到直到直到直到直到直到直到直到直到直到直到直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直达方法。