A hybrid Decoder-DeepONet operator regression framework for unaligned observation data

Deep neural operators (DNOs) have been utilized to approximate nonlinear mappings between function spaces. However, DNOs are confronted with challenges stemming from expanded dimensionality and computational costs tied to unaligned observation data, which ultimately compromise the accuracy of predictions. In this study, we present a hybrid Decoder-DeepONet framework to effectively handle unaligned data. This framework is advanced through its extension to the Multi-Decoder-DeepONet, which leverages an average field to enhance input augmentation. Furthermore, on the basis of the universal approximation theorem, we demonstrate that these frameworks preserve consistencies with operator approximation theory despite the substitution of the product with a decoder net. Two numerical experiments, Darcy problem and flow-field around an airfoil, are conducted to demonstrate the advantages of the proposed methods over conventional DeepONet approaches. The results reveal that both Decoder-DeepONet and Multi-Decoder-DeepONet utilize more compact training data dimensions and occupy less space, markedly enhancing prediction accuracy in the context of unaligned data.