  Viscous shocks are a particular type of extreme events in nonlinear
multiscale systems, and their representation requires small scales. Model
reduction can thus play an important role in reducing the computational cost
for an efficient prediction of shocks. Yet, reduced models typically aim to
approximate large-scale dominating dynamics, which do not resolve the small
scales by design. To resolve this representation barrier, we introduce a new
qualitative characterization of the space-time locations of shocks, named as
the ``shock trace'', via a space-time indicator function based on an empirical
resolution-adaptive threshold. Different from the exact shocks, the shock
traces can be captured within the representation capacity of the large scales,
which facilitates the forecast of the timing and locations of the shocks
utilizing reduced models. Within the context of a viscous stochastic Burgers
equation, we show that a data-driven reduced model, in the form of nonlinear
autoregression (NAR) time series models, can accurately predict the random
shock traces, with relatively low rates of false predictions. The NAR model
significantly outperforms the corresponding Galerkin truncated model in the
scenario of either noiseless or noisy observations. The results illustrate the
importance of the data-driven closure terms in the NAR model, which account for
the effects of the unresolved small scale dynamics on the resolved ones due to
nonlinear interactions.
