An analysis of the derivative-free loss method for solving PDEs
Jihun Han Yoonsang Lee
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.