This work proposes a solution for the problem of training physics informed
networks under partial integro-differential equations. These equations require
infinite or a large number of neural evaluations to construct a single residual
for training. As a result, accurate evaluation may be impractical, and we show
that naive approximations at replacing these integrals with unbiased estimates
lead to biased loss functions and solutions. To overcome this bias, we
investigate three types of solutions: the deterministic sampling approach, the
double-sampling trick, and the delayed target method. We consider three classes
of PDEs for benchmarking; one defining a Poisson problem with singular charges
and weak solutions, another involving weak solutions on electro-magnetic fields
and a Maxwell equation, and a third one defining a Smoluchowski coagulation
problem. Our numerical results confirm the existence of the aforementioned bias
in practice, and also show that our proposed delayed target approach can lead
to accurate solutions with comparable quality to ones estimated with a large
number of samples. Our implementation is open-source and available at
this https URL

本文提出了针对基于部分积分 - 微分方程训练物理启发网络的问题的解决方案，包括三种类型的解决方法，证实了当前的近似方法的存在偏差，提出了新的延迟目标方法，证明其可以以可比的质量获得精确的解决方案。