Reconstructing original data points from t-SNE output
There is no closed form.

It is a local embedding, and you really cannot expect to find a good inverse mapping.

It's doable, but not very good. Use gradient descent in the input domain. Just like tSNE but with the t-distribution and the Gaussian reversed. However, the original Gaussian has a sigma parameter that you also need to find. So you need to also optimize this to have the desired perplexity. You'll need to do the math for this optimization problem yourself.