In addition to what written by @enricoanderlini, I would also point out that there don't seem to be many datapoints in your test set. Having a (small) number of points, exposes you, even more, to the impact of leverage / influential points (you can also read about it here). With regards to this, @enricoanderlini is correct when he suggests cross-validation.
Interestingly, while the focus is often on model capacity (degrees of freedom), your set-up highlights that ANN calibration, in a train-validate-test pipeline, is "hungrier" of datapoints than, say, just a OLS with a train-test split.
The results seem to suggest overfitting is a problem at the moment.
Have you tried to select the training, validation and test sets differently? What I mean is, choose a number of different sets, say 5-10, from the whole data set you have. This way if you observe the same behaviour with all of them you know you have to redesign the NN.
If overfitting is a problem, you may want to decrease the number of neurons.