How to Remove the Patch Artifacts of Neural Network Denoising Process?
One simple way to solve it is using Overlapping Patches.

Let's say you have image which is $ 20 \times 20 $ and you work on patches of the size $ 5 \times 5 $.
As I understand from your description you do 16 times denoising of $ 5 \times 5 $ patches.

What you should do is run the patches mask like in convolution. So each pixels (Ignoring boundaries) will be filtered in 25 different patches.  

Now you have 25 different estimates for each pixel.
The trivial solution is to use their mean.
More advanced methods will use some prior on how to create a good weighted combination as output.
One prior could be keeping the composition of the image as smooth as possible.
Other methods could be built around estimation of the noise in each patch.