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.

翻译：深度神经算子（DNOs）已被用于逼近函数空间之间的非线性映射。然而，DNOs面临非对齐观测数据带来的维度增加和计算成本挑战。本研究提出一种混合Decoder-DeepONet算子回归框架以有效处理非对齐数据。此外，我们引入了一种多解码器深度神经算子（Multi-Decoder-DeepONet），该算子利用训练数据的平均场进行输入增强。基于通用逼近定理，证明了该框架与算子逼近理论的一致性。通过达西问题和翼型绕流场两个数值实验验证了所提方法的效率与精度。结果表明，Decoder-DeepONet和Multi-Decoder-DeepONet在处理非对齐观测数据方面具有优势，并展示了其在提升预测精度方面的潜力。