Physics-Informed Neural Networks (PINNs) have gained popularity in scientific computing in recent years. However, they often fail to achieve the same level of accuracy as classical methods in solving differential equations. In this paper, we identify two sources of this issue in the case of Cauchy problems: the use of $L^2$ residuals as objective functions and the approximation gap of neural networks. We show that minimizing the sum of $L^2$ residual and initial condition error is not sufficient to guarantee the true solution, as this loss function does not capture the underlying dynamics. Additionally, neural networks are not capable of capturing singularities in the solutions due to the non-compactness of their image sets. This, in turn, influences the existence of global minima and the regularity of the network. We demonstrate that when the global minimum does not exist, machine precision becomes the predominant source of achievable error in practice. We also present numerical experiments in support of our theoretical claims.

翻译：物理信息神经网络（Physics-Informed Neural Networks, PINNs）近年来在科学计算领域备受关注。然而，在求解微分方程时，它们往往无法达到传统方法同等水平的精度。本文针对柯西问题，识别了导致这一问题的两个根源：使用$L^2$残差作为目标函数，以及神经网络的逼近间隙。我们证明，仅最小化$L^2$残差与初始条件误差之和并不足以保证真实解，因为该损失函数无法捕捉潜在动力学特性。此外，由于神经网络像集非紧致，它们无法捕捉解中的奇异性。这进而影响全局最小值的存在性及网络的正则性。我们证明，当全局最小值不存在时，机器精度成为实际可达误差的主要来源。我们还通过数值实验支持了理论论断。