  In this paper, we propose a DeepONet structure with causality to represent
the causal linear operators between Banach spaces of time-dependent signals.
The theorem of universal approximations to nonlinear operators proposed in
\cite{tianpingchen1995} is extended to operators with causalities, and the
proposed Causality-DeepONet implements the physical causality in its framework.
The proposed Causality-DeepONet considers causality (the state of the system at
the current time is not affected by that of the future, but only by its current
state and past history) and uses a convolution-type weight in its design. To
demonstrate its effectiveness in handling the causal response of a physical
system, the Causality-DeepONet is applied to learn the operator representing
the response of a building due to earthquake ground accelerations. Extensive
numerical tests and comparisons with some existing variants of DeepONet are
carried out, and the Causality-DeepONet clearly shows its unique capability to
learn the retarded dynamic responses of the seismic response operator with good
accuracy.
