I have a problem that I've framed out in a particular way, but I don't know if I'm re-inventing the wheel here. Is there an existing literature base in this problem? Does it have a corresponding term of art? Where would you recommend that I start reading about this type of problem?
The basic problem that I have is that I need to make predictions about where ants will be in a large terrain. I have all sorts of information about the terrain -- its elevation, bodies of water, the flora cover, the distances to cities and other human settlements, the weather -- all of these things plausibly influence where and how ants move across this terrain.
In addition to the terrain data, I have data about where ants have historically been found with lat/longs to a high degree of precision.
We also know that ants will generally be moving from one end of the terrain to the other. But there is not a hard requirement that they always move closer to the opposite end at every step. We just know that they're starting at one end and want to go somewhere on the other end. The "box" of terrain is just a very wide rectangle, and I'm comfortable with assuming that ants want to go from the longest side to the opposite longest side.
Plausibly some ant could go from one corner to the farthest corner, but it's more likely that they'll go to a nearer point because the path is shorter, so probably the path is less costly. This is where the calculus of variations comes in: I want to make inferences about the plausible paths that ants will take. Likewise, an ant could make a several U-turns to avoid some obstacle (like a suddenly steep elevation, or crossing a body of water), and doing so would incur a cost cost, but probably won't make several such U-turns, if judicious planning could avoid it ahead of time by taking an alternative route. This is similar to how water will usually flow down hill, but special circumstances can make a geyser.
Conceptually, the cost function is a function of the terrain data, but the cost itself is unknown.
The straight-up machine learning approach would just be to use the terrain data to make inferences about where ants will be directly as some sort of regression problem -- this can be done any number of arbitrary ways, but I wonder if doing so without considering paths will lead to implausible inferences, such as making predictions that ants might be at any point along a line, except some vast swath in the middle where the model predicts probability 0, or that there is a "dotted line" of alternating high- and low-probability regions. I wouldn't really believe that model, and would tend to think that it's overfitting the data. It seems like making inferences about the cost function, however indirectly, would impose a certain amount of regularization on the solution driven purely by the "pure" machine learning regression problem, so incorporating that information would improve our inferences.
So where would I start reading about this style of problem?