Two-dimensional MSE loss in MLP training


I want to train a multilayer perceptron that maps a multidimensional input to a two-dimensional target. The target consists of two different variables (gross primary productivity and evapotranspiration) that are also of very different magnitude: mean of the first is ~ 0.4, mean of the second ~ -22 . Doing so, I construct a fully-connected model with two output dimensions and linear activation, which I evaluate in the MSE-loss criterion against my batch sample with the same two output dimensions.

One variable is predicted pretty good - but the other is predicted completely wrong, following the shape of the first variable. I scaled the target in order to reduce the effect of the different units, but the results remain.
The batch loss is small at first, and then increasing to a fixed loss - just the opposite as I expect it.

Is this approach with a two-dimensional target reasonable at all, will I have to use a different loss function or a completely different approach overall? Maybe just use two different MLPs, one for each target?

I am very happy about any comments or shared experiences on this! Thank you.