High-fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics

In computational fluid dynamics, high-fidelity simulations are often computationally expensive, requiring significant CPU time and hardware resources. This paper addresses the challenge of creating reduced-order models (ROMs) that can accurately mirror the behavior of complex physical processes without requiring intrusive Galerkin projections of the governing equations. The author seeks to develop a digital twin data model (DTM) that is both computationally efficient and highly accurate.

To achieve this, the author proposes a two-stage framework:

1. Offline Stage: The author employs randomized dynamic mode decomposition (DMD) to identify a reduced-order subspace. By using randomized singular value decomposition (RSVD), the algorithm avoids the computational bottleneck of standard SVD, allowing for faster identification of the most influential (leading) dynamic modes.
2. Online Stage: The author uses a cascade-forward neural network as a Nonlinear Autoregressive Estimator (NLARX) to model the temporal evolution of the DMD coefficients. This allows the DTM to predict the system's dynamics over time with minimal computational overhead.

The framework is tested on the nonlinear viscous Burgers equation, simulating shock wave phenomena at increasing Reynolds numbers (Re = 10^2, 10^3, and 10^4).

The proposed DTM framework successfully captures the dynamics of shock wave phenomena with high precision. The randomized DMD approach effectively selects the optimal number of leading modes, and the NLARX estimator provides accurate temporal predictions. The study demonstrates that the DTMs achieve excellent correlation coefficients with the original numerical data while significantly reducing the computational burden compared to full-order simulations. The offline stage is particularly efficient, requiring less than two seconds of CPU time, while the online stage provides rapid, accurate predictions even for complex, high-Reynolds-number scenarios.