  Areas of computational mechanics such as uncertainty quantification and
optimization usually involve repeated evaluation of numerical models that
represent the behavior of engineering systems. In the case of complex nonlinear
systems however, these models tend to be expensive to evaluate, making
surrogate models quite valuable. Artificial neural networks approximate systems
very well by taking advantage of the inherent information of its given training
data. In this context, this paper investigates the improvement of the training
process by including sensitivity information, which are partial derivatives
w.r.t. inputs, as outlined by Sobolev training. In computational mechanics,
sensitivities can be applied to neural networks by expanding the training loss
function with additional loss terms, thereby improving training convergence
resulting in lower generalisation error. This improvement is shown in two
examples of linear and non-linear material behavior. More specifically, the
Sobolev designed loss function is expanded with residual weights adjusting the
effect of each loss on the training step. Residual weighting is the given
scaling to the different training data, which in this case are response and
sensitivities. These residual weights are optimized by an adaptive scheme,
whereby varying objective functions are explored, with some showing
improvements in accuracy and precision of the general training convergence.
