Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering problems. The mathematical understanding of these methods is limited, and in particular, it seems that, a consistent notion of stability is missing. Towards addressing this issue we consider model problems of partial differential equations, namely linear elliptic and parabolic PDEs. We consider problems with different stability properties, and problems with time discrete training. Motivated by tools of nonlinear calculus of variations we systematically show that coercivity of the energies and associated compactness provide the right framework for stability. For time discrete training we show that if these properties fail to hold then methods may become unstable. Furthermore, using tools of $\Gamma-$convergence we provide new convergence results for weak solutions by only requiring that the neural network spaces are chosen to have suitable approximation properties.

翻译：物理信息神经网络是一种利用神经网络逼近偏微分方程解的数值方法，该方法已获得广泛关注，目前被应用于大量物理与工程问题中。然而，这类方法的数学理解仍存在局限，尤其缺乏一致的稳定性概念。为应对这一问题，我们以偏微分方程的模型问题——即线性椭圆型与抛物型偏微分方程为研究对象，考察了具有不同稳定性特征的问题以及时间离散训练问题。受非线性变分学工具启发，我们系统论证了能量强制性与相关紧致性为稳定性提供了恰当框架。对时间离散训练而言，研究表明若这些性质不成立，则方法可能变得不稳定。进一步地，利用Γ-收敛工具，我们仅需假设神经网络空间具有适当的逼近性质，即可获得弱解的新收敛性结论。