When you try to discover governing equations from data, the first problem isn't fitting. It's the search space. A generic polynomial basis for a PDE with several variables and derivatives generates hundreds or thousands of candidate terms. Most are physically meaningless. Fitting to all of them guarantees overfitting. The standard approach is to try all terms and let sparsity-promoting regression discard the ones that don't matter. This works, but it's computationally expensive and doesn't know why terms are absent.
Yokokura and Takeuchi invert the process. Instead of starting with everything and pruning, they start with the system's symmetries and derive which terms are permitted. Symmetry generators act as linear operators on the space of candidate terms. The kernel of those operators โ the terms that are invariant under the symmetries โ is exactly the allowed search space. They demonstrate this on Toner-Tu equations for active matter and KPZ growth equations, recovering the known forms and identifying higher-order extensions.
The structural point: the constraint does more work than the data. When you know what symmetries a system has, the symmetries alone eliminate most of the search space before any measurement is taken. The data's job reduces from "find the equation" to "determine the coefficients" โ a dramatically simpler problem.
This is the opposite of brute-force discovery. Instead of asking which terms the data supports, you ask which terms the symmetries permit. The permitted list is provably complete, so nothing is missed. And everything not on the list is structurally impossible, so nothing irrelevant contaminates the fit. The symmetry is doing the thinking.