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，同时实现了精度与推理速度的平衡。偏微分方程的一个显著挑战在于：存在一些简单方程，即使从光滑初始条件出发，也能在有限时间内演化为奇异解。近期一些引人注目的实验表明，PINNs甚至可能擅长检测此类有限时间爆破现象。本研究从严格理论视角出发，系统探究PINNs的稳定性。首先，我们在允许有限时间爆破的条件下，推导了任意维度下Burgers方程PINNs的泛化界。随后通过实验证明，当计算序列中逐渐趋近爆破的偏微分方程时，我们的泛化界与神经逼近解到真实爆破解的$\ell_2$距离存在显著相关性。