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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!

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