  We consider the problem of online scheduling on a single machine in order to
minimize weighted flow time. The existing algorithms for this problem (STOC
'01, SODA '03, FOCS '18) all require exact knowledge of the processing time of
each job. This assumption is crucial, as even a slight perturbation of the
processing time would lead to polynomial competitive ratio. However, this
assumption very rarely holds in real-life scenarios.
  In this paper, we present the first algorithm for weighted flow time which do
not require exact knowledge of the processing times of jobs. Specifically, we
introduce the Scheduling with Predicted Processing Time (SPPT) problem, where
the algorithm is given a prediction for the processing time of each job,
instead of its real processing time. For the case of a constant factor
distortion between the predictions and the real processing time, our algorithms
match all the best known competitiveness bounds for weighted flow time --
namely $O(\log P), O(\log D)$ and $O(\log W)$, where $P,D,W$ are the maximum
ratios of processing times, densities, and weights, respectively. For larger
errors, the competitiveness of our algorithms degrades gracefully.
