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.

该研究分析了使用神经网络解决一类椭圆型偏微分方程的无导数损失方法。该方法利用费曼 - 卡克公式，结合随机游走和其相应的平均值。我们研究了费曼 - 卡克公式中与时间间隔相关的效果，以及与计算效率、可训练性和采样误差的关系。我们的分析表明，训练损失的偏差与时间间隔和神经网络的空间梯度成正比，与游走大小成反比。我们还表明，时间间隔必须足够长以训练网络。这些分析结果告诉我们可以根据时间间隔的最优下限尽可能选择较小的游走大小。我们还提供数值测试来支持我们的分析。

求解偏微分方程之无导数损失方法之分析

An analysis of the derivative-free loss method for solving PDEs

We use tools from $n$-dimensional Brownian motion in conjunction with the
Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact
Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that
any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim
\frac{1}{\sqrt{\lambda}}$. This also gives a slight refinement of a result by
Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other
hand, we also prove that no nodal domain can be contained in a reasonably thin
tubular neighbourhood of unions of finitely many surfaces inside $M$.

利用 n 维布朗运动中的工具结合热扩散的 Feynman-Kac 公式，研究紧致黎曼流形 M 上的节点几何。一方面，我们通过扩展 Lieb 定理证明任何节点域 Omega^lambda 几乎完全包含半径约为 1 / 根号下 lambda 的球，从而略微完善了 Mangoubi 的结果。另一方面，我们还证明没有节点域可以被包含在 M 内有限多个曲面的合理狭窄的管状邻域内。