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Does an engine using a vector field exist?
What I mean is that instead of calculating intersections, for example one could calculate the vector field generated by gravity and for the ground and let the object follow those fields.
What are the advantages and drawbacks of this kind of game engine?
The more vectors you have, the more you have to store, and the more floating-point operations you (potentially) have to do. I have no doubt that this method would require a vast number of vectors. Why instance vectors that may never be used? It wouldn't make sense from am efficient space/time usage perspective.
Intersection checks cannot possibly be replaced by your proposed field physics. In classical mechanics, at a human scale, objects will not noticeably affect each other by gravitic forces in any case. The only time they are going to "push away from one another" is when they are intersecting in your simulation, then and only then. Thus there is no way a field-based approach could eliminate the need to perform intersection checks (in whatever form).
The discrepancy between what you are imagining and the reality of computer simulations, is as follows:
A simulation runs through some finite number of steps per time interval (whether these steps are variable or equal duration is irrelevant). Based on this, objects must "jump" a certain distance in each time step -- this process is not analog -- based on their velocity, no matter how small. So logically there are always going to be cases where an object may potentially "jump into" and intersect another, hence the need for intersection checks.
The real world, on the other hand, doesn't work in the same way that computer simulations do, because real-world time is (in practice) analog. So the machine that is the universe, so to speak, resolves collisions in a different manner, using something like the approach you propose. Atoms never really collide, they just apply forces that grow stronger and stronger the closer you force them together. That means that objects, which are made of atoms, obey similar rules: forces are applied on a continual, analog basis between every object in existence -- so the physicists tell us.
P.S. Subject to the eminent Martin Sojka's proofreading for errors of a physical or mathematical nature. ;)
