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 https://github.com/ehsansaleh/btspinn.

翻译：本文针对物理知识嵌入网络在部分积分-微分方程训练过程中存在的问题提出了解决方案。这类方程需要无限次或大量神经评估才能构建单个训练残差，导致精确评估难以实现。研究表明，若简单使用无偏估计替代这些积分，将产生有偏损失函数和偏差解。为消除此偏差，我们研究了三类解决方案：确定性采样方法、双重采样技巧以及延迟目标法。基准测试涵盖三类偏微分方程：包含奇点电荷与弱解的泊松问题、涉及电磁场弱解与麦克斯韦方程的算例，以及描述斯莫卢霍夫斯基凝聚过程的第三类问题。数值结果不仅证实了上述偏差在实际场景中的存在性，还表明我们提出的延迟目标法能够获得与大量采样评估质量相当的精确解。本实现的源代码已开源，详见 https://github.com/ehsansaleh/btspinn。