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

Area of Science:

• Nonlinear dynamics
• Computational physics
• Data-driven modeling

Background:

• Viscous shocks are extreme events in nonlinear multiscale systems requiring small-scale resolution.
• Model reduction is crucial for computational efficiency in shock prediction but struggles with small scales.
• A representation barrier exists for reduced models in capturing shock dynamics.

Purpose of the Study:

• Introduce a new method,
• shock trace
• , to characterize space-time shock locations.
• Enable prediction of shock timing and locations using reduced models.
• Overcome the limitations of reduced models in resolving small-scale dynamics.

Main Methods:

• Developed a space-time indicator function for shock trace identification using an empirical, resolution-adaptive threshold.
• Utilized nonlinear autoregression (NAR) time series models as a data-driven reduced model.
• Compared NAR model performance against a Galerkin truncated model.

Main Results:

• Shock traces can be captured by large scales, facilitating prediction with reduced models.
• NAR models accurately predicted random shock traces with low false prediction rates.
• NAR models with data-driven closure terms significantly outperformed Galerkin models.

Conclusions:

• Data-driven closures in NAR models are vital for approximating unresolved small-scale dynamics.
• The shock trace concept effectively bridges the gap between reduced models and extreme event prediction.
• This approach enhances the predictive capability of reduced models for nonlinear multiscale systems.