论文标题
关于近似PDE的物理知情神经网络（PINN）的概括误差的估计值
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
论文作者
论文摘要
物理知情的神经网络（PINN）最近被广泛用于PDE的稳健和准确近似。我们在近似PDE的正向问题的PINNS的概括误差上提供了严格的上限。引入了抽象的形式主义，并利用了基础PDE的稳定性特性，以根据训练误差和训练样本的数量来得出对概括错误的估计。该抽象框架用非线性PDE的几个示例进行了说明。还提出了验证所提出的理论的数值实验。
Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.
