  Deep neural operators can learn nonlinear mappings between
infinite-dimensional function spaces via deep neural networks. As promising
surrogate solvers of partial differential equations (PDEs) for real-time
prediction, deep neural operators such as deep operator networks (DeepONets)
provide a new simulation paradigm in science and engineering. Pure data-driven
neural operators and deep learning models, in general, are usually limited to
interpolation scenarios, where new predictions utilize inputs within the
support of the training set. However, in the inference stage of real-world
applications, the input may lie outside the support, i.e., extrapolation is
required, which may result to large errors and unavoidable failure of deep
learning models. Here, we address this challenge of extrapolation for deep
neural operators. First, we systematically investigate the extrapolation
behavior of DeepONets by quantifying the extrapolation complexity via the
2-Wasserstein distance between two function spaces and propose a new behavior
of bias-variance trade-off for extrapolation with respect to model capacity.
Subsequently, we develop a complete workflow, including extrapolation
determination, and we propose five reliable learning methods that guarantee a
safe prediction under extrapolation by requiring additional information -- the
governing PDEs of the system or sparse new observations. The proposed methods
are based on either fine-tuning a pre-trained DeepONet or multifidelity
learning. We demonstrate the effectiveness of the proposed framework for
various types of parametric PDEs. Our systematic comparisons provide practical
guidelines for selecting a proper extrapolation method depending on the
available information, desired accuracy, and required inference speed.
