Tensor decompositions, a generalization of matrix factorizations to higher dimensions, have proven to be very effective for data reduction. For scientific data, which are inherently high- dimensional, tensor decompositions are able to exploit naturally occurring higher-order interactions to discover low-rank formats, achieving orders of magnitude reduction for only moderate loss of accuracy. However, the traditional formats typically preserve accuracy with respect to a global tensor norm, which may be insufficient for preserving various scientific quantities of interest (QoIs) or invariants (e.g. conservation laws). To address this shortcoming, we developed goal-oriented tensor decompositions [1] that aim to minimize the discrepancy between tensor model data and original data as well as functions of tensor model data and functions of original data. This is achieved by augmenting the traditional Frobenius- norm loss term of the objective function with additional terms that represent loss of accuracy of QoIs that are functions of the data, a concept similar to Physics-Informed Neural Networks (PINNs). This talk will present the goal-oriented formulation for two tensor formats—the Canonical Polyadic (CP) and the Tucker decompositions—and techniques for solving the resulting optimization problem using trust-region methods and Gauss-Newton Hessian approximations. Results of applying the goal-oriented formulation will be presented for two exemplar high-fidelity simulation data sets: turbulent combustion and plasma physics. Across the two cases, a slew of conserved QoIs were considered—total mass, momentum, kinetic energy, internal energy, magnetic energy—and the results show that, in comparison to traditional formulations, the goal-oriented formulations are able to significantly reduce the error in these quantities with no significant loss of accuracy in terms of the tensor Frobenius- norm.