Solving the inverse problem to find a hidden object

25 Mar, 2026

I was listening to StarTalk and Neil deGrasse Tyson told the story of the discovery of Neptune, which first happened mathematically and then observationally!

Essentially, the effect of Neptune on the trajectories of other planets caused discrepancies that shouldn't have been there. This clued-in two mathematicians of the 19th century, John C. Adams and Urbain Le Verrier, who looked into what could be causing those discrepancies.

A toy problem

We can set up equivalent inverse problem. Imagine a spring mass system which could have one or two identical masses (meaning they have the same weight). One spring connects the first mass to the wall. If another mass exists, then an identical spring connects it to the first mass. The masses can only move left and right and let's say there is no friction.

A friend of yours perturbs the system and records the position of everything, but only gives you the position of the first mass, $x_{1} (t)$ for all $t \geq 0$ . They ask:

if there's a second mass and
can you predict where the second mass is.

As far as classical mechanics exercises go, this one is fairly straightforward. The system follows some well-known law if we only have one mass and we can easily simulate that law by using the known $x_{1} (0)$ , producing $\hat{x_{1}} (t)$ . Then we can compare $x_{1} (t)$ with $\hat{x_{1}} (t)$ for $t > 0$ .

If there's good agreement, then we know with some confidence that there is a single mass.
Otherwise, we will see a discrepancy! In the single-mass case, $x_{1} (t)$ is a single-frequency oscillation. With two masses, it becomes a superposition of two frequencies.

This is exactly the same logic as in the discovery of Neptune: when a model of the world fails to explain observations, the discrepancy itself contains information about what is missing.

You don’t observe Neptune directly. You observe its effect, and work backwards.

However, you don't have to imagine all this, as I asked Claude to make a public demonstration that I think makes sense.