论文标题
完全非线性偏微分方程的深部分支求解器
A deep branching solver for fully nonlinear partial differential equations
论文作者
论文摘要
我们提出了用于完全非线性PDE的数值解的随机分支算法的多维深度学习实现。这种方法旨在通过将神经网络的使用与蒙特卡洛分支算法相结合，以应对涉及任何订单的梯度项的功能非线性。与其他深度学习PDE求解器相比，它还使我们能够检查学习的神经网络功能的一致性。提出的数值实验表明，该算法可以根据后向随机微分方程或Galerkin方法胜过深度学习方法，并提供在完全非线性示例中这些方法获得的溶液估计值。
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.
