Particular solutions, or solutions in general, can be thought of as a combination of waves within a finite period of time. The creation of the waves are generally constrained by limits of physics of oscillators. We also typically associate certain types of solutions with number of particles and certain types of oscillators.
Due to the definable finiteness, it would seem that one could create a mathematical space that was sufficiently large such that each conceivable solution produced by oscillators in the universe would be defined as a point in that space. So in some wild way, it is in fact the determination and mapping of the space of solutions that that creates the solutions, and it is merely up to a group of scientists to later discover the points of interest in that space.
Not being a scientist myself, what sort of physical parameters and variables would I consider to define the space of solutions in physics?