This study aims to utilize a deep operator network (DeepONet) to estimate the aortic pressure waveform generated by a 1D hemodynamic model of the human arterial system when provided with a given input flow rate waveform. The arterial system model employed in this research is a modified version of the previously validated ADAN56 model. It is known that changes in the shape, magnitude, and period of the incoming flow rate waveform to the aorta lead to variations in the aortic pressure waveform. Realistic flow rate waveforms were generated based on previous human-based studies, and corresponding aortic pressure waveforms in the arterial system model were calculated using the open-source hemodynamic solver openBF. The branch network of the DeepONet receives a single cycle of the flow rate waveform, while the trunk network receives a normalized time point and heart rate of input flow rate waveform. This utilizes the periodic nature of the signals in both the flow rate and pressure waveforms. The test set RMSE was 0.337 ± 0.086 mmHg. The time required for inferring a single pressure waveform was reduced from an average of 57.02 seconds using the hemodynamic solver to an average of 0.095 seconds using the proposed network. This suggests the capability of our network to rapidly and accurately estimate aortic pressure waveforms based on flow rate waveforms.