The synoptic scale variability of the Indian summer monsoon (ISM) is contributed by the weak cyclonic vortices known as low-pressure systems (LPSs). LPSs are the primary mechanism by which central Indian plains receive rainfall. Traditionally, synoptic variability is considered to have a low predictability. In the present study, we developed a framework, namely, LPS Neural Operator (LPSNO), using the neural operator-based deep learning to predict the spatial structure of daily mean sea level pressure anomalies over the Bay of Bengal at a resolution of 1°x1°. The proposed neural operator extends the Fourier neural operator framework by employing convolutional LSTMs in the operator backbone. Further, the mean sea level pressure is reconstructed using the predicted anomaly and the climatology, which is then used to track the LPSs using a Lagrangian tracking algorithm. The median pattern correlation between the predicted and actual mean sea-level pressure anomalies over the BoB is about 88%, 60%, and 50% for 24, 48, and 72-hour forecasts, respectively. The proposed model improves the accuracy of predictions compared with the earlier ConvLSTM models. The pattern correlation between the observed and predicted synoptic activity index (SAI) is 0.94, 0.9, and 0.87 for 1, 2, and 3-day ahead predictions, respectively. A well-trained model of LPSNO takes only ~3.2 s to generate a one-day forecast on a single GPU node of Nvidia V100, which is computationally extremely cheap compared to the conventional numerical weather prediction models. The proposed LPSNO can advance operational weather forecasting substantially.