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, a high-dimensional Poisson's equation, an interface problem, and an application to a chemotaxis population model.

我们介绍了一种基于深度神经网络的方法来求解一类椭圆型偏微分方程。我们用一个深度神经网络来逼近PDE的解决方案，该神经网络是根据Feynman-Kac公式的精神在PDE的概率表示的指导下训练的。该解决方案是由布朗运动驱动的mar过程的期望给出的。随着布朗步行者探索领域，深度神经网络将使用强化学习的形式进行迭代训练。我们的方法是“无导数损失法”，因为它不需要为了计算训练损失而相对于输入神经元显式计算神经网络的导数。我们的方法的优点在一系列测试问题中得到了展示：拐角奇点问题，