The differential equations steering motion in classical mechanics are very famous: $$\sum_i\frac{{\bf F}_i}{m}={\bf a} = \frac{d{\bf v}(t)}{dt} = \frac{d{\bf x}(t)^2}{dt^2}$$
Now to the question, can we given an estimate of positions over time calculate "backwards" which sources of forces are affecting each object?
Lately (last 10 years at least) it seems popular to do mixed norm optimization, and $L_1$ seems to give us sparse solutions which we want if we want as few sources of forces (simple description) as possible.
Say we have a "dictionary" of forces regularized with $L_1$ and the forces have parametric descriptions ( for example polynomials or splines or whatever ).
What would make this mathematically different or "more difficult" to pose and solve such problems than any dictionary learning method in any of the other fields where it has proven to be very useful (so far)?
If this is already done, please point me in any direction where I can read more about it.