A neural network system includes a first neural network configured to predict a mean value output and epistemic uncertainty of the output given input data, and a second neural network configured to predict total uncertainty of the output of the first neural network. The second neural network is trained to predict total uncertainty of the output of the first neural network given the input data through a training process involving minimizing a cost function that involves differences between a predicted mean value of a geophysical property of a geological formation from the first neural network and a ground-truth value of the geophysical property of the geological formation. The neural network system further includes one or more processors configured to run a software module that determines aleatoric uncertainty of the output of the first neural network based on the epistemic uncertainty of the output and the total uncertainty of the output.