Grid-Free Approach to Partial Differential Equations on Volumetric Domains [pdf]

Just from the title I knew Keenan Crane was the advisor. He has some really good YouTube video explanations for most of the concepts:

https://www.youtube.com/watch?v=bZbuKOxH71o

This work is one of those things that feels like the completely obviously right way to do things in retrospect, and why hasn't anyone implemented this before? It helps that the authors explain it very intuitively, slowly building up the tools to run like stars.

Something I don't see mentioned is that this would be pretty useful for training a physics-based model, specifically a neural ODE. Since the mean squared error scales as O(1/N), to converge you need O(1/Loss) evaluations per point. If you were using a grid approach, the cell size would need to be O(sqrt(Loss)) in size, and the running time couldn't be better than O(1/(Loss * log(Loss))) in two dimensions. To be fair, it probably takes O(1/log(Loss)) time for each Monte Carlo simulation, so it's no worse to use the grid. But, if you go up to three or more dimensions, this method still has the same running time while the best grid method takes at least O(# grid cells) = O(1/Loss^(d/2)) time.

This touches on some ideas I toyed with in the past around switching DEs to an integral representation and throwing optimization / energy minimization at them. Combining this with Monte Carlo integration is a nut I thought about, but never really cracked, but it's interesting to see that this type of thing is already done in in modern rendering approaches. Will definitely be spending some time this weekend looking at this.

What I'm interested in seeing if this could be used for multidisciplinary design optimization. Currently it is possible to do coupled aero structural optimization of wings and the like but of course you have to deal with all the mesh stuff and make it deformable as wing changes shape. And then get the gradients out.

Very curious to see if it might be possible to greatly simplify the problem with this method

I just saw this and I'm surprised that it hasn't gotten more feedback- this is sensational. I'm just working my way through the paper as a Humanities major and I find it advanced enough that you're clearly treading new water but it is accessible enough that I don't feel like a pleb. Great work! Enjoy your PhD doctor!