[measurablefunc]

How do they prove their model preserves conservation principles? I looked in the paper & didn't find any evidence of how they verify that whatever their "trained" model is doing is actually physically plausible & maintains the relevant invariants like mass, energy, momentum, etc.

[flwi]

Author here: we do NOT do conservation of energy/momentum. We are currently trying to incorporate that in a follow up paper, but for now, the models that try that (e.g. PINNs (soft constraint) or hard constraint models, all perform badly when modeling multiple systems.

Perhaps, we will encounter the bitter lesson again and a well trained model will solve this. But as I said, we are also looking at hybrid models

[bobmarleybiceps]

I think very few of these "replace numerical solver with ML model" papers do anything to verify invariants are satisfied (they often are not well preserved). They basically all just check that the model approximately reproduces some dynamics on a test data of PDEs, that's often sampled from the same distribution as the training dataset...

[flwi]

Author here: we do NOT do conservation of energy/momentum. We are currently trying to incorporate that in a follow up paper, but for now, the models that try that (e.g. PINNs (soft constraint) or hard constraint models, all perform bad when modeling multiple systems.

Perhaps, we will encounter the bitter lesson again and a well trained model will solve this. But as I said, we are also looking at hybrid models

[woctordho]

I guess it can be implemented in the 'sampler' part. When solving an actual PDE, project the output of the AI onto a space that preserves the invariants.

[esafak]

From a quick scan, I do not think they explicitly encode that. They want "the model to predict the evolution of diverse physical systems governed by partial differential equations". It looks like a more sophisticated sibling of time series forecasting models rather than a physics-informed nonparametric symbolic regression model.

[NeoInHacker]

Yeah, It’s true that PDEs are the "top-tier tool" for describing physical phenomena—from the laws of motion in classical mechanics and electromagnetic waves in electromagnetism to the evolution of wave functions in quantum mechanics, they accurately model most macroscopic, classical scenarios. However, when it comes to covering all physical phenomena, they really "fall short": in quantum gravity, spacetime may be discontinuous, making the concept of differentiation meaningless; for complex systems like turbulence, PDEs cannot be solved nor can they capture macroscopic laws; even for the randomness of quantum measurements, PDEs can only predict probability distributions and fail to explain the underlying nature. In short, they are a "top-tier auxiliary," but by no means a "one-size-fits-all key."

[codethief]

> in quantum gravity

GP was asking about conservation laws but in gravity you don't even have energy-momentum conservation.

[ogogmad]

Why? Is this important as a sanity check in the absence of any independent verifications?

[bobmarleybiceps]

I'm not an expert on this, so take this with a grain of salt. Chaotic PDEs are extremely sensitive to initial conditions. This essentially makes it so that any numerical solution will (quickly) diverge from the true solution over time. (Just due to floating point error, discretization error, etc.) This is why for a lot of turbulent navier-stokes stuff, people don't necessarily care about the specific phenomena that occur, but look at statistical properties.

I think one of the reasons it is important to preserve conservation laws is that, at the very least, you can be confident that your solution satisfies whatever physical laws your PDE relies on, even if it's almost certainly not the "actual" solution to the PDE. You actually can ensure that a numerical solver will approximately satisfy conservation laws. Then at the very least, even if your solution diverges from the "actual" PDEs solution, you can have some confidence that it's still a useful exploration of possible states. If conservation laws are not preserved AND your solution diverges from the "actual" PDE solution, then you probably cannot be confident about the model's utility.

[bobmarleybiceps]

Actually I just happened to see this: https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how.... It's basically explaining the same thing, but much better than me :-)