I stumbled upon the notion of a universal differential equation.
A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.
Here is an example UDE:
$$\frac{d^4y}{dt^4} \frac{dy}{dt}^2 - 3 \frac{d^3y}{dt^3} \frac{d^2y}{dt^2} \frac{dy}{dt} + 2(1 - n^{-2})\frac{d^2y}{dt^2}^3 = 0$$ $$\forall n > 3$$
This result is theoretically interesting to me, but it isn't clear to me how this can be used to help study a continuous function. Given a continuous function $f:\mathbb{R} \mapsto \mathbb{R}$ that I would like to approximate with $y$, what might a UDE such as the one above help me learn about $f$?