Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the $\ell_2$-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

翻译：物理信息神经网络（PINNs）在数值求解复杂偏微分方程方面不断取得新突破，同时提供了精度与推理速度之间的理想权衡。偏微分方程的一个特别挑战在于，存在一些简单的偏微分方程，从光滑初始条件出发可能在有限时间内演化为奇异解。近期一些显著实验表明，PINNs甚至可能擅长检测此类有限时间爆破现象。本研究旨在从严格理论视角系统探究PINNs的稳定性。首先，我们推导了在允许有限时间爆破条件下任意维度Burgers偏微分方程PINNs解的泛化界。随后通过实验证明，当计算序列中越来越接近爆破的偏微分方程时，我们的泛化界与神经网络替代解到真实爆破解的$\ell_2$距离显著相关。