As I read this, I get stuck on the form of the solutions they wish to solve. For example, in the lava examples, presumably at a static time snapshot, the mathematicians wish to generate a function of space point in; temperature out. Then, maybe, do this at multiple time points, or evolve the system over time. Or maybe generalize classes of how a lava system could evolve.

This is a very complicated model of the real world, and I think this sort of problem comes up whenever we move from "spherical cow" physics and math to modeling or simulating something? There's chaos in the system, and sensitivity to initial conditions which aren't known. It's like reading about the "3 body problem is unsolvable".

Maybe you look at this without the framework of PDEs, and simulate it. But the article implies that lava is heterogeneous, so you don't know how to model how each part of it interacts with the rest. I struggle understanding, for example, how the author uses the word "equation" to describe something this complicated.

So, maybe the ideal solution is a set of coarse descriptions of the lava flow's temperature distribution, likelyhoods for each, predictions of which you get depending on how much you know about the initial conditions. Probably fractal?

It's often the case that describing how a complex system changes with its input variables is much easier than writing the function from the variable to the state.

A PDE is a precise description of some unknown function in terms of how it changes, so it's really the ideal framework for doing the kind of simulation you're talking about.

PDE can simulate Kelvin helmhotlz instability and if you want to go even smaller, you can go to particle in cell methods. And the distribution thing you are talking about is similar to lattice boltzmann methods.