  For many systems in science and engineering, the governing differential
equation is either not known or known in an approximate sense. Analyses and
design of such systems are governed by data collected from the field and/or
laboratory experiments. This challenging scenario is further worsened when
data-collection is expensive and time-consuming. To address this issue, this
paper presents a novel multi-fidelity physics informed deep neural network
(MF-PIDNN). The framework proposed is particularly suitable when the physics of
the problem is known in an approximate sense (low-fidelity physics) and only a
few high-fidelity data are available. MF-PIDNN blends physics informed and
data-driven deep learning techniques by using the concept of transfer learning.
The approximate governing equation is first used to train a low-fidelity
physics informed deep neural network. This is followed by transfer learning
where the low-fidelity model is updated by using the available high-fidelity
data. MF-PIDNN is able to encode useful information on the physics of the
problem from the {\it approximate} governing differential equation and hence,
provides accurate prediction even in zones with no data. Additionally, no
low-fidelity data is required for training this model. Applicability and
utility of MF-PIDNN are illustrated in solving four benchmark reliability
analysis problems. Case studies to illustrate interesting features of the
proposed approach are also presented.
