In mathematics, asking about an “explicit formula” for solutions to problems means just: Assuming a general background in mathematics, is the solution something I already have spent years of my life developing an intuition for?
Having spent some time looking at numerical solutions, I would generalize this to “do we have an algorithm that allows us to efficiently explore the relevant domain space?”
The algorithm could be a way to calculate sin(x), to square a number, to solve the Laplace equation with given boundary conditions, or even to calculate the distribution of gravitational radiation from black hole collisions. As long as it as efficient (in terms of computational and human efforts) as pressing sin(x) on the calculator, it is as good as exact.
Having spent some time looking at numerical solutions, I would generalize this to “do we have an algorithm that allows us to efficiently explore the relevant domain space?”
The algorithm could be a way to calculate sin(x), to square a number, to solve the Laplace equation with given boundary conditions, or even to calculate the distribution of gravitational radiation from black hole collisions. As long as it as efficient (in terms of computational and human efforts) as pressing sin(x) on the calculator, it is as good as exact.