[riskneutral]

> It’s all good to write down approximate solutions to some hard n body problem or non linear pde, just don’t look too far ahead in time, or update your model with fresh data if you need to.

It sounds like this symbolic regression approach can potentially find correct PDE solutions directly from data:

"In February, they fed their system 30 years’ worth of real positions of the solar system’s planets and moons in the sky. The algorithm skipped Kepler’s laws altogether, directly inferring Newton’s law of gravitation and the masses of the planets and moons to boot."

Note that Newton’s law of gravitation can also be derived from a differential equation. Also note that simply knowing the exact solution to the PDE does not mean that the PDE model itself is an exact representation of reality. For example, we know from Einstein's laws that Newton's laws are not correct at astronomical scales. In the symbolic regression approach, the solution would be limited by the number of different observable variables available to train the model on.

Many PDEs in real world problems also do not have any known exact symbolic solution, so approximate symbolic formulas are used routinely anyway but are just painstakingly discovered and derived by humans with "deeper understanding" (actually, just hoards of PhD students and their advisors throwing everything they can think of at the problem until one of them lands on a unusable formula).

[mrow84]

> In the symbolic regression approach, the solution would be limited by the number of different observable variables available to train the model on.

What do you mean here? Aren't all approaches limited by the number of observables available (except speculatively)?

[riskneutral]

Well, with a human coming up with a PDE model (i.e. a system of equations) based on their intuition about what the causes of a given phenomenon are, they would be limited by their intuition/imagination about which variables are significant for the model. For all practical purposes, the variables that a modeler imagines to be significant are a subset of all the possible observable variables. A machine learning approach would learn the significant variable directly from the data, and is therefore not limited to imagination. There's an example of this mentioned in the article, in the context of ocean models:

“The algorithm picked up on additional terms,” Zanna said, producing a “beautiful” equation that “really represents some of the key properties of ocean currents, which are stretching, shearing and [rotating].”