  A multi-fidelity regression model is proposed for combining multiple datasets
with different fidelities, particularly abundant low-fidelity data and scarce
high-fidelity observations. The model builds upon recent multi-fidelity
frameworks based on neural networks, which employ two distinct networks for
learning low- and high-fidelity data, and extends them by feeding the gradients
information of low-fidelity data into the second network, while the gradients
are computed using automatic differentiation with minimal computational
overhead. The accuracy of the proposed framework is demonstrated through a
variety of benchmark examples, and it is shown that the proposed model performs
better than conventional multi-fidelity neural network models that do not use
gradient information. Additionally, the proposed model is applied to the
challenging case of turbulent flow reconstruction. In particular, we study the
effectiveness of the model in reconstructing the instantaneous velocity field
of the decaying of homogeneous isotropic turbulence given
low-resolution/low-fidelity data as well as small amount of
high-resolution/high-fidelity data. The results indicate that the proposed
model is able to reconstruct turbulent field and capture small scale structures
with good accuracy, making it suitable for more practical applications.
