I derived a lower bound which only depends on moments (e.g. mean and variance).
Even if the true distribution is unkown, we can calculate the lower bound (approximation) of the KL-divergence using only the expected value and the variance of a function we choose.
Please see Theorem 1 in the following paper.
https://arxiv.org/abs/1907.00288
URL of a sample code for this paper is
https://github.com/nissy220/KL_divergence
Please confirm the results.
In the left graph, the red line is the result of our lower bound and the blue line is the KL-divergence for the normal distribution.
The right graph displays the ratio.