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.

研究以理论角度验证 PINNs 的稳定性，推导了 Burgers' PDE 通用界限，实验证明界限与由神经网络找到的虚拟爆破解与真实爆破解之间的 L2 距离密切相关。