  Deep Operator Network (DeepONet) is a neural network framework for learning
nonlinear operators such as those from ordinary differential equations (ODEs)
describing complex systems. Multiple-input deep neural operators (MIONet)
extended DeepONet to allow multiple input functions in different Banach spaces.
MIONet offers flexibility in training dataset grid spacing, without constraints
on output location. However, it requires offline inputs and cannot handle
varying sequence lengths in testing datasets, limiting its real-time
application in dynamic complex systems. This work redesigns MIONet, integrating
Long Short Term Memory (LSTM) to learn neural operators from time-dependent
data. This approach overcomes data discretization constraints and harnesses
LSTM's capability with variable-length, real-time data. Factors affecting
learning performance, like algorithm extrapolation ability are presented. The
framework is enhanced with uncertainty quantification through a novel Bayesian
method, sampling from MIONet parameter distributions. Consequently, we develop
the B-LSTM-MIONet, incorporating LSTM's temporal strengths with Bayesian
robustness, resulting in a more precise and reliable model for noisy datasets.
