A Comparison of Parametric Dynamic Mode Decomposition Algorithms for Thermal-Hydraulics Applications

TL;DR

This work evaluates three parametric Dynamic Mode Decomposition (pDMD) strategies—Reduced Operators Interpolation (ROI), Reduced Koopman Operator Interpolation (RKOI), and Interpolation of Latent Dynamics (Monolithic and Partitioned variants)—on thermal-hydraulics problems ranging from laminar flow over a cylinder to a RELAP5-based DYNASTY facility. By combining reduced-order representations via SVD with parameter regression, the study highlights trade-offs between offline preprocessing and online forecasting accuracy across complex, parametric time-series data. ROI offers computational speed but may miss high-frequency features in complex flows, while RKOI, leveraging Optimised DMD, generally provides better accuracy with favorable online costs; Latent-Dynamics approaches deliver high fidelity at the expense of online complexity. Overall, no single method universally outperforms the others; the choice depends on dynamics, required forecast horizon, and available computational resources, with future work aimed at data assimilation and multi-physics applications.

Abstract

In recent years, algorithms aiming at learning models from available data have become quite popular due to two factors: 1) the significant developments in Artificial Intelligence techniques and 2) the availability of large amounts of data. Nevertheless, this topic has already been addressed by methodologies belonging to the Reduced Order Modelling framework, of which perhaps the most famous equation-free technique is Dynamic Mode Decomposition. This algorithm aims to learn the best linear model that represents the physical phenomena described by a time series dataset: its output is a best state operator of the underlying dynamical system that can be used, in principle, to advance the original dataset in time even beyond its span. However, in its standard formulation, this technique cannot deal with parametric time series, meaning that a different linear model has to be derived for each parameter realization. Research on this is ongoing, and some versions of a parametric Dynamic Mode Decomposition already exist. This work contributes to this research field by comparing the different algorithms presently deployed and assessing their advantages and shortcomings compared to each other. To this aim, three different thermal-hydraulics problems are considered: two benchmark 'flow over cylinder' test cases at diverse Reynolds numbers, whose datasets are, respectively, obtained with the FEniCS finite element solver and retrieved from the CFDbench dataset, and the DYNASTY experimental facility operating at Politecnico di Milano, which studies the natural circulation established by internally heated fluids for Generation IV nuclear applications, whose dataset was generated using the RELAP5 nodal solver.

Figures (18)

• Figure 1: Train and Test split of the Reynolds number for the Flow Over Cylinder solved with dolfinx.
• Figure 2: Decay of the singular values (top left), contour plots of the first 4 SVD modes of velocity $\mathbf{u}$ (top right), underlining the hierarchical spatial features and temporal evolution of the latent dynamics for fixed Reynolds (bottom). The singular values show an exponential decay, highlighting the fact that low rank modes retain the majority of the information content.
• Figure 3: Contour plots of the velocity magnitude for the test parameters at the final time of the simulation, comparing the pDMD versions with the full-order solution (FOM).
• Figure 4: Train and Test split of the Reynolds number for the Flow Over Cylinder from the CFDbench dataset luo_cfdbench_2024.
• Figure 5: Average (with respect to test parameters) relative errors over time for the different pDMD versions applied on the CFDbench dataset.