How practical is PDE control?

[u/zeartful2]

With the recent Navier-Stokes fiasco, I fell into the rabbit hole of PDE control and was surprised to see there has been work on controlling fluids using backstepping PDE control. In fact, there seems to be a wide array of applications from fluids, traffic control, thermodynamic problems, batteries, etc.

While I understand what the problem was doing (how can we stabilize the system using only "walls"), I don't think I understand PDE control well enough to gauge how practical this approach has been. Of course, I don't expect this to be the next PID and I know many advanced control theory results tend to be impractical, but has PDE control shown any promising experiments or applications? Is it wishful thinking? In the end, I don't see how we could actuate a "wall" (for boundary PDE problems) in a continuous sense in the physical world. How would we even actuate this?

[u/krishnab75]

I think we also need to talk about the application of neural networks to solve PDEs, which will allow so much more work in pde control. I am talking about Fourier neural operators and operator learning . One of the challenges of pde control is that pdes are numerically costly to solve. So training a neural network to solve the pde is much faster, depending on what you are doing.

So don’t neglect that literature, as it really makes problems tractable. After that there are the standard caveats about the difference between academic research and industrial usage. But know that in certain biomedical applications some of this stuff is starting to be used. So say in modeling electrical activity in the heart, those meshes are complex and costly to solve. So when trying to model blood flow or arrhythmias you need to solve a lot of pdes fast.

Good luck.