We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples.

翻译：我们提出了一种用于完全非线性偏微分方程数值求解的随机分支算法的多维深度学习实现。该方法通过将神经网络与蒙特卡洛分支算法相结合，旨在处理涉及任意阶梯度项的函数非线性。与其他深度学习PDE求解器相比，该方法还允许我们检验所学习神经网络函数的一致性。数值实验表明，该算法能超越基于倒向随机微分方程或伽辽金法的深度学习方法，并在完全非线性示例中提供这些方法无法获得的解估计。