  Deep operator networks (DeepONets) are receiving increased attention thanks
to their demonstrated capability to approximate nonlinear operators between
infinite-dimensional Banach spaces. However, despite their remarkable early
promise, they typically require large training data-sets consisting of paired
input-output observations which may be expensive to obtain, while their
predictions may not be consistent with the underlying physical principles that
generated the observed data. In this work, we propose a novel model class
coined as physics-informed DeepONets, which introduces an effective
regularization mechanism for biasing the outputs of DeepOnet models towards
ensuring physical consistency. This is accomplished by leveraging automatic
differentiation to impose the underlying physical laws via soft penalty
constraints during model training. We demonstrate that this simple, yet
remarkably effective extension can not only yield a significant improvement in
the predictive accuracy of DeepOnets, but also greatly reduce the need for
large training data-sets. To this end, a remarkable observation is that
physics-informed DeepONets are capable of solving parametric partial
differential equations (PDEs) without any paired input-output observations,
except for a set of given initial or boundary conditions. We illustrate the
effectiveness of the proposed framework through a series of comprehensive
numerical studies across various types of PDEs. Strikingly, a trained physics
informed DeepOnet model can predict the solution of $\mathcal{O}(10^3)$
time-dependent PDEs in a fraction of a second -- up to three orders of
magnitude faster compared a conventional PDE solver. The data and code
accompanying this manuscript are publicly available at
\url{https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets}.
