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.

翻译：本研究针对部分积分微分方程约束下的物理信息神经网络训练问题提出解决方案。此类方程需要无限次或大量神经网络评估才能构建用于训练的单个残差项，导致精确计算在实际中难以实现。我们证明，若采用无偏估计简单替代这些积分，将导致损失函数及解产生系统性偏差。为克服此偏差，我们探究了三类潜在解决方案：确定性采样方法、双采样技巧及延迟目标法。我们选用三类偏微分方程进行基准测试：第一类定义具有奇异电荷的泊松问题及高达10维的弱解；第二类涉及电磁场弱解及麦克斯韦方程；第三类定义斯莫鲁霍夫斯基凝聚问题。数值结果证实了上述偏差在实际中的存在性，并表明我们提出的延迟目标法能够获得与大规模采样积分估计结果质量相当的精确解。本研究的开源实现位于 https://github.com/ehsansaleh/btspinn。