  This work proposes a solution for the problem of training physics-informed
networks under partial integro-differential equations. These equations require
an 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 potential solutions: the deterministic sampling
approaches, the double-sampling trick, and the delayed target method. We
consider three classes of PDEs for benchmarking; one defining Poisson problems
with singular charges and weak solutions of up to 10 dimensions, 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 sample size integral. Our
implementation is open-source and available at
https://github.com/ehsansaleh/btspinn.
