This study analyzes the derivative-free loss method to solve a certain class of elliptic PDEs using neural networks. The derivative-free loss method uses the Feynman-Kac formulation, incorporating stochastic walkers and their corresponding average values. We investigate the effect of the time interval related to the Feynman-Kac formulation and the walker size in the context of computational efficiency, trainability, and sampling errors. Our analysis shows that the training loss bias is proportional to the time interval and the spatial gradient of the neural network while inversely proportional to the walker size. We also show that the time interval must be sufficiently long to train the network. These analytic results tell that we can choose the walker size as small as possible based on the optimal lower bound of the time interval. We also provide numerical tests supporting our analysis.

翻译：本研究分析了使用神经网络求解一类椭圆型偏微分方程的无导数损失方法。该方法基于Feynman-Kac公式，通过引入随机行走粒子及其对应的平均值来实现。我们探究了与Feynman-Kac公式相关的时间步长和粒子数量对计算效率、可训练性和采样误差的影响。分析表明：训练损失偏差与时间步长和神经网络的空间梯度成正比，与粒子数量成反比。同时，我们发现时间步长必须足够长才能有效训练网络。这些解析结果表明，可根据时间步长最优下界选择尽可能小的粒子数量。我们还提供了支持理论分析的数值实验验证。