If You Had o4-Level Models, What Previously Impossible Research Would Become Possible?

Over the past couple of days I went to UT Austin to visit a friend, and I sat in on a few physics classes with him.

One of them was on numerical solutions to general relativity. The class was taught by a very energetic woman.

For the entire class, she was telling us just how complicated Einstein’s equations are.

Analytical solutions? Except for a tiny handful of special cases, do not even think about them.

For numerical solutions, you first have to prove that the whole PDE is well posed – I do not really know what that means, but it seems to be something like small changes in the inputs leading to small changes in the solution – and then find an initial condition before you can even start computing.

That does not sound too complicated, but for a very concrete case, humanity had to wait until 1950 to get this far for the first time.

And for more complicated cases, today’s methods are basically one method per situation, with no generality to speak of.

As a machine learning student doing pseudo-science plus pseudo-math plus pseudo-engineering, I only half understood it. After class I wanted to ask a question, but I could not come up with anything genuinely physics-related, so I had to ask:

“If AI breaks through in the next couple of years, say o4-o5-level models appear, and they can reason about mathematical physics at the level of an ordinary mathematician, then if I gave you ten billion mediocre mathematicians like that, what previously unsolvable problems could now be solved?”

I originally thought she would say something like: AI could classify all kinds of edge cases in much finer detail, then gradually build up a macroscopic picture, similar to how, when the four-color theorem was solved, computers divided all possible topological cases into billions of categories and checked them one by one.

But her answer was more interesting:

“Right now, the numerical methods used to solve Einstein’s equations are very slow and not very accurate. Most of them use first-order finite difference methods to estimate the derivatives in the partial differential equations.

“The main reason is that physicists are already overwhelmed by all the mathematical prerequisites.

“For first-order differences alone, there are ten formulas: the first four are used to determine a snapshot solution in time, meaning the spatial state at a particular instant, and the remaining six are used for time evolution.

“If you want to use a higher-order algorithm such as Runge-Kutta, deriving the expansion alone might take mathematicians hundreds of lifetimes, and humans cannot afford that derivation. So no such algorithm exists right now.

“With AI, though, expanding numerical methods for PDEs should be pure labor and should be automatable. A higher-order algorithm designed this way might have tens of thousands of lines of formulas and hundreds of thousands of terms, but its accuracy and convergence speed would both be much better.”

That answer hit me hard.

We know that improvements in intelligence mainly come from scaling: more data and more compute. Pretraining, post-training, and the newly popular test-time compute all work this way.

But this was the first time I realized that scaling can also be applied to the order of a Taylor expansion.

It turns out people use low-order approximations simply because they are being lazy – or rather, because they cannot calculate any further and it is too exhausting.

In the future, with AI, it will not matter how many lines of formulas there are.

At that point, even if there is only a tiny gain in precision, as long as there is no constraint, expand it to twelfth order and see what happens; let it throw an error before saying the theory has a problem.

Thinking one step further. Before AI, intelligence was expensive; in the future, it will be cheap. Without AI, if figuring something out clearly did not create at least tens of thousands of dollars of value, nobody would hire a PhD to study it properly. In the future, even if something only has a few cents of value, we should still ask AI to calculate it clearly, because the electricity costs only a few cents.

So everyone should broaden their imagination. Beyond the order of Taylor expansion, there must be many other pure-labor things in the future that automation can push to the extreme. It is very interesting to think about.

Feel the AGI!

Update:

What I want to discuss here is this: if we can pile up a huge amount of simple intelligence, what kind of qualitative change, or emergent capability, can that quantitative change produce?

As an analogy, imagine that we are Euler or Gauss, sitting there by hand calculating all kinds of ODEs.

In that traditional domain, modern people are not necessarily stronger than those two.

At this point, suppose we obtain a computer that can perform an enormous number of simple calculations.

The trivial, obvious idea is that we can do all kinds of arithmetic with large and small numbers, no longer needing to copy out a table of logarithms from 1 to 100000 by hand. Analytical solutions can also be calculated more accurately.

However, the non-trivial, non-obvious idea is that for ODEs where no analytical solution can be found, we can directly brute-force a numerical solution.

For people in the nineteenth century, because the cost was too high, that idea would have been rejected by consciousness the moment it appeared in the subconscious; it would never even have had a chance to enter serious discussion.

They were even less likely to imagine that this technology could become cheap enough to use in a video game’s physics engine – further beyond them still, how could they have understood?

This is one kind of qualitative change produced by quantitative change.

So let us imagine: if future AI can perform an enormous amount of simple mathematical reasoning, what can it do that could not be done before?

The obvious idea is that it can automate the mental labor humans already do today.

But the non-obvious ideas are more interesting to discuss: the operations we subconsciously reject because they are too expensive.

This post is trying to propose one such dimension: the complexity of numerical methods.

Likewise, in the future there should be many more ways of using ultra-cheap intelligence that are still beyond our current imagination.