  Physics-informed neural networks (PINNs) have recently emerged as a novel and
popular approach for solving forward and inverse problems involving partial
differential equations (PDEs). However, achieving stable training and obtaining
correct results remain a challenge in many cases, often attributed to the
ill-conditioning of PINNs. Nonetheless, further analysis is still lacking,
severely limiting the progress and applications of PINNs in complex engineering
problems. Drawing inspiration from the ill-conditioning analysis in traditional
numerical methods, we establish a connection between the ill-conditioning of
PINNs and the ill-conditioning of the Jacobian matrix of the PDE system.
Specifically, for any given PDE system, we construct its controlled system.
This controlled system allows for adjustment of the condition number of the
Jacobian matrix while retaining the same solution as the original system. Our
numerical findings suggest that the ill-conditioning observed in PINNs
predominantly stems from the Jacobian matrix. As the condition number of the
Jacobian matrix decreases, PINNs exhibit faster convergence rates and higher
accuracy. Building upon this understanding and the natural extension of
controlled systems, we present a general approach to mitigate the
ill-conditioning of PINNs, leading to successful simulations of the
three-dimensional flow around the M6 wing at a Reynolds number of 5,000. To the
best of our knowledge, this is the first time that PINNs have been successful
in simulating such complex systems, offering a promising new technique for
addressing industrial complexity problems. Our findings also offer valuable
insights guiding the future development of PINNs.
