Physics Informed Neural Networks (PINNs) have recently gained popularity for solving partial differential equations, given the fact they escape the curse of dimensionality. In this paper, we present Physics Informed Neural Networks as an underdetermined point matching collocation method then expose the connection between Galerkin Least Square (GALS) and PINNs, to develop an a priori error estimate, in the context of elliptic problems. In particular, techniques that belong to the realm of least square finite elements and Rademacher complexity analysis are used to obtain the error estimate.
翻译:最近,物理信息神经网络(PINNs)在解决部分差异方程式方面获得了受欢迎的呼声,因为它们逃脱了维度的诅咒。 在本文中,我们将物理信息神经网络作为与同地安置方法相匹配的不确定点,然后暴露了Galerkin最低广场(GALS)和PINNs之间的联系,以便结合椭圆问题,制定一个先验的误差估计。 特别是,使用了属于最不平的定点元素领域的技术和Rademacher复杂程度分析来获得误差估计。