  Transfer learning (TL) enables the transfer of knowledge gained in learning
to perform one task (source) to a related but different task (target), hence
addressing the expense of data acquisition and labeling, potential
computational power limitations, and dataset distribution mismatches. We
propose a new TL framework for task-specific learning (functional regression in
partial differential equations (PDEs)) under conditional shift based on the
deep operator network (DeepONet). Task-specific operator learning is
accomplished by fine-tuning task-specific layers of the target DeepONet using a
hybrid loss function that allows for the matching of individual target samples
while also preserving the global properties of the conditional distribution of
target data. Inspired by the conditional embedding operator theory, we minimize
the statistical distance between labeled target data and the surrogate
prediction on unlabeled target data by embedding conditional distributions onto
a reproducing kernel Hilbert space. We demonstrate the advantages of our
approach for various TL scenarios involving nonlinear PDEs under diverse
conditions due to shift in the geometric domain and model dynamics. Our TL
framework enables fast and efficient learning of heterogeneous tasks despite
significant differences between the source and target domains.
