Tensor Train Representation of High-Dimensional Unsteady Flamelet Manifolds

Abstract

This study, for the first time, investigates the use of tensor trains (TTs) to represent high-dimensional unsteady flamelet progress variable (UFPV) manifolds in chemically reacting computational fluid dynamics (CFD). The UFPV framework captures the thermochemical state of reacting flows using a reduced set of parameters and pre-computed manifolds, avoiding the need to transport all species or solve large stiff reaction systems. High-dimensional manifolds enhance accuracy by resolving coupled thermochemical effects critical in high-speed reacting flows but impose substantial memory demands. Here, a five-dimensional UFPV manifold is constructed and stored in the TT format to address this limitation. Several chemical mechanisms and table sizes are examined to evaluate TT compression performance and accuracy. The TT representation achieves significant memory reduction while preserving manifold fidelity and combustion behavior. A one-dimensional reacting-flow case using the discontinuous Galerkin (DG)-based JENRE Multiphysics Framework confirms that TT-compressed manifolds are interchangeable with standard UFPV tables. In addition to memory reduction, benchmark tests show that TT-based manifold sampling can achieve up to 2.4X speedup relative to dense tensor evaluation. Although demonstrated for UFPV combustion models, the proposed TT framework is broadly applicable to other tabulation-based combustion methodologies and provides a scalable alternative to machine learning (ML)-based approaches for representing high-dimensional combustion manifolds.

Figures (6)

• Figure 1: 5D UFPV implementation in the JENRE® Multiphysics Framework.
• Figure 2: Schematic illustration of the tensor train (TT) decomposition of a $d$--dimensional table. Each TT core $C^{(k)} \in \mathbb{R}^{r_{k-1} \times N_k \times r_k}$ is a third--order tensor, where $N_k$ denotes the size of the $k$th table dimension and $r_k$ are the TT ranks connecting neighboring cores, with boundary ranks fixed as $r_0 = r_d = 1$. The arrows labeled $i_k$ indicate the physical indices of the original table, which select slices of each TT core during tensor evaluation, while the internal rank indices $r_k$ are contracted to reconstruct the tensor value.
• Figure 3: Relationship between tensor train (TT) compression and $L^2$ error for all species in the GRI 3.0 mechanism. Each color corresponds to a different tolerance, with the indicated mean compression factor.
• Figure 4: Histogram of memory usage for a 22-species ethylene mechanism comparing a 5D UFPV table with its tensor train (TT) representation. The UFPV table requires $\sim$1.5 GB, whereas TT needs only $\sim$14.6 MB ($\sim$103$\times$ total compression). Red numbers denote per-species compression factors. Species bars with red diagonal hatch patterns denote minor species (tolerance 0.05), while solid-filled bars indicate major species (tolerance 0.01).
• Figure 5: Comparison of UFPV and TT results for a 1D periodic flame for the selected species mass fractions ($Y_{H_2}$, $Y_{O_2}$, $Y_{H_2O}$, $Y_{OH}$) as functions of streamwise coordinate $x$.