# Recent research on solving "inverse" ODE problems with neural networks?

I come from a physics background, and I am not familiar with the state-of-the-art research in solving ODE optimization problems with NNs.

Let me briefly introduce this so-called "inverse" problem. For the following ODE, $$f_2(t) \frac{d^2 x}{dt^2} + f_1(t) \frac{dx}{dt} + f_0(t) x = g(t)$$ I would like the solution to be similar to a "target" function as much as possible, i.e., I may consider minimizing the loss function $$|x(t) - x^{\text{target}}(t)|, \forall t \in[t_{i},t_{f}]$$, where $$[t_{i},t_{f}]$$ is the time interval of interest. With what kind of $$\{f_2(t),f_1(t),f_0(t),g(t)\}$$, can this ODE produce the best solution $$x(t)$$?

I wonder if there is any recent work on this kind of problem? Any NN-based (or optimization-based) methods, papers, codes/libraries are welcomed!

Thanks!