Having a knowledge of the mathematics helps.
An example is the Rydberg equation which was empirically derived in the late 19th century and describes the energy level spacings versus wavelength λ in the spectrum of the hydrogen atom.
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Rₕ is Rydberg’s constant for the hydrogen atom, n₁ and n₂ are known as the principal quantum numbers where n₂ > n₁.
Scientists had very little idea at the time of how the empirical relationship worked or why Rₕ had a value of 1.097×10⁷ m⁻¹.
When the quantum mechanical model for the hydrogen atom was created in the 1920s the Schrödinger equation was found to be:
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On solving this equation, it was found the equation could be reduced to a simpler form of the type.
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The solutions for this equation are known as associated Laguerre polynomials.
Remarkably these polynomials lead to the Rydberg equation which could now be explained in terms of the quantum mechanical model of the hydrogen atom.
This leads to a philosophical issue was the Rydberg equation invented or discovered?
The methods used in the 19th century would indicate it was an invention but in the 1920s it was derived from the principles of quantum mechanics and was a discovery.
Is the Rydberg equation like mathematics in general already 'out there' waiting to be discovered or is it a product of the human mind?
There is also the case raised by the Nobel Prize winner in physics Eugene Wigner that the laws of nature are written in the language of mathematics.