Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp
This work addresses the stability of PINNs for challenging PDEs with singularities, which is important for computational physics and numerical methods, but it is incremental as it builds on existing PINN research.
The authors investigated the ability of Physics Informed Neural Networks (PINNs) to solve Burgers' PDE near finite-time blow-up, deriving generalization bounds and showing experimental correlation with the distance from the true blow-up solution.
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.