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Shock trace prediction by reduced models for a viscous stochastic Burgers equation

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.