Basically approximating a solution, and then finding an equation from some data. Nothing wrong with writing down an approximation to some problem. An approximation can be very useful. As long as you know it’s limitations it’s fine to answer questions where it applies. But this can not replace deeper understanding. 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’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).
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].”
Yes I would say most of physics we currently can make predictions with are just effective models that are only applicable at some certain scale. Certainly there may be some underlying ideas that get shared around but doing calculations and making predictions is effective.
Yes, they are in absolute terms. To prove that they are exact we would need to have a complete understanding of the universe, which we do not have.
In practical terms, some of them are in the sense that it is impossible to measure their inaccuracy with common equipment and our primitive human senses. In the end, their purpose is to make predictions and help us make sense of what’s around us, they do not need to be exact.
The ones you learn in high school are, but general and special relativity are accurate to trillionths of a percent (their approximations at low speed and without time dilation are the high school equations).
We don’t know if they break down, because we have no means of measuring them more accurately on earth with all the statistical fluctuations happening.
I think the point is that with conventional equations we can make increasingly better approximations by just using pen and paper, while with these ML methods, you need to train your system ab initio if you want to improve your approximations.
