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Deep Operator Network (DeepONet) is a neural network framework for learningnonlinear 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 constraintson output location. However, it requires offline inputs and cannot handlevarying sequence lengths in testing datasets, limiting its real-timeapplication in dynamic complex systems. This work redesigns MIONet, integratingLong Short Term Memory (LSTM) to learn neural operators from time-dependentdata. This approach overcomes data discretization constraints and harnessesLSTM's capability with variable-length, real-time data. Factors affectinglearning performance, like algorithm extrapolation ability are presented. Theframework is enhanced with uncertainty quantification through a novel Bayesianmethod, sampling from MIONet parameter distributions. Consequently, we developthe B-LSTM-MIONet, incorporating LSTM's temporal strengths with Bayesianrobustness, resulting in a more precise and reliable model for noisy datasets.

B-LSTM-MIONet: Bayesian LSTM-based Neural Operators for Learning the  Response of Complex Dynamical Systems to Length-Variant Multiple Input  Functions