论文标题
一种无衍生的方法，用于用深神经网络求解椭圆形偏微分方程
A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networks
论文作者
论文摘要
我们引入了一种基于神经网络的深度方法，用于求解一类椭圆形偏微分方程。我们使用深层神经网络近似PDE的解决方案，该溶液是在feynman-kac公式的精神的概率代表的指导下训练的。该解决方案是由对布朗尼运动驱动的Martingale过程的期望给出的。当布朗步行者探索该领域时，深层神经网络是使用一种强化学习形式进行迭代训练的。我们的方法是一种“无衍生损耗方法”，因为它不需要针对输入神经元的神经网络的衍生物进行明确计算以计算训练损失。我们方法的优点在一系列测试问题中展示了：角落奇异性问题，界面问题以及趋化性人群模型的应用。
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic representation of the PDE in the spirit of the Feynman-Kac formula. The solution is given by an expectation of a martingale process driven by a Brownian motion. As Brownian walkers explore the domain, the deep neural network is iteratively trained using a form of reinforcement learning. Our method is a 'Derivative-Free Loss Method' since it does not require the explicit calculation of the derivatives of the neural network with respect to the input neurons in order to compute the training loss. The advantages of our method are showcased in a series of test problems: a corner singularity problem, an interface problem, and an application to a chemotaxis population model.
