  The well-known governing physics in science and engineering is often based on
certain assumptions and approximations. Therefore, analyses and designs carried
out based on these equations are also approximate. The emergence of data-driven
models has, to a certain degree, addressed this challenge; however, the purely
data-driven models often (a) lack interpretability, (b) are data-hungry, and
(c) do not generalize beyond the training window. Operator learning has
recently been proposed as a potential alternative to address the aforementioned
challenges; however, the challenges are still persistent. We here argue that
one of the possible solutions resides in data-physics fusion, where the
data-driven model is used to correct/identify the missing physics. To that end,
we propose a novel Differentiable Physics Augmented Wavelet Neural Operator
(DPA-WNO). The proposed DPA-WNO blends a differentiable physics solver with the
Wavelet Neural Operator (WNO), where the role of WNO is to model the missing
physics. This empowers the proposed framework to exploit the capability of WNO
to learn from data while retaining the interpretability and generalizability
associated with physics-based solvers. We illustrate the applicability of the
proposed approach in solving time-dependent uncertainty quantification problems
due to randomness in the initial condition. Four benchmark uncertainty
quantification and reliability analysis examples from various fields of science
and engineering are solved using the proposed approach. The results presented
illustrate interesting features of the proposed approach.
