  Deep neural operators (DNOs) have been utilized to approximate nonlinear
mappings between function spaces. However, DNOs face the challenge of increased
dimensionality and computational cost associated with unaligned observation
data. In this study, we propose a hybrid Decoder-DeepONet operator regression
framework to handle unaligned data effectively. Additionally, we introduce a
Multi-Decoder-DeepONet, which utilizes an average field of training data as
input augmentation. The consistencies of the frameworks with the operator
approximation theory are provided, on the basis of the universal approximation
theorem. Two numerical experiments, Darcy problem and flow-field around an
airfoil, are conducted to validate the efficiency and accuracy of the proposed
methods. Results illustrate the advantages of Decoder-DeepONet and
Multi-Decoder-DeepONet in handling unaligned observation data and showcase
their potentials in improving prediction accuracy.
