Learning-Augmented Performance Model for Tensor Product Factorization in High-Order FEM

TL;DR

This work addresses the challenge of predicting performance for arithmetic-intensive tensor-product factorization kernels in high-order FEM, where traditional memory-bandwidth models fail to capture instruction-level effects. It introduces a dependency-chain–based analytical model that links loop-body splitting to FMA instruction dependencies, and enhances it with a learning-augmented component using XGBoost to predict the key dependency ratio. Evaluations on Fujitsu A64FX and Intel Xeon Gold 6230 show the hybrid model achieves substantially lower errors (MAPE roughly 1%–24% on A64FX and 1%–13% on Xeon, with some degradation at edge cases) compared with Roofline and ECM across polynomial orders. The approach offers a systematic, interpretable framework for tuning loop-body splitting in high-order FEM tensor kernels and has potential applicability to broader arithmetic-intensive HPC kernels.

Abstract

Accurate performance prediction is essential for optimizing scientific applications on modern high-performance computing (HPC) architectures. Widely used performance models primarily focus on cache and memory bandwidth, which is suitable for many memory-bound workloads. However, it is unsuitable for highly arithmetic intensive cases such as the sum-factorization with tensor $n$-mode product kernels, which are an optimization technique for high-order finite element methods (FEM). On processors with relatively high single instruction multiple data (SIMD) instruction latency, such as the Fujitsu A64FX, the performance of these kernels is strongly influenced by loop-body splitting strategies. Memory-bandwidth-oriented models are therefore not appropriate for evaluating these splitting configurations, and a model that directly reflects instruction-level efficiency is required. To address this need, we develop a dependency-chain-based analytical formulation that links loop-splitting configurations to instruction dependencies in the tensor $n$-mode product kernel. We further use XGBoost to estimate key parameters in the analytical model that are difficult to model explicitly. Evaluations show that the learning-augmented model outperforms the widely used standard Roofline and Execution-Cache-Memory (ECM) models. On the Fujitsu A64FX processor, the learning-augmented model achieves mean absolute percentage errors (MAPE) between 1% and 24% for polynomial orders ($P$) from 1 to 15. In comparison, the standard Roofline and ECM models yield errors of 42%-256% and 5%-117%, respectively. On the Intel Xeon Gold 6230 processor, the learning-augmented model achieves MAPE values from 1% to 13% for $P$=1 to $P$=14, and 24% at $P$=15. In contrast, the standard Roofline and ECM models produce errors of 1%-73% and 8%-112% for $P$=1 to $P$=15, respectively.

Figures (14)

• Figure 1: Illustration of sum factorization on hexahedral element.
• Figure 2: Execution time of the original and fast modal filtering (sum factorization implemented) at $P=7$ and $P=11$ren_implementing_2024.
• Figure 3: Execution time of the fast modal filtering (sum factorization implemented) at $P=8$ in different splitting configurations on the Fujitsu A64FX processor ren_performance_2025.
• Figure 4: Illustration of floating point arithmetic pipeline status with and without data dependency.
• Figure 5: Illustration of mapping loop body splitting configurations to the FMA instruction dependency chains.