GPU-accelerated ECSW hyper-reduction: implementation and performance evaluation

Abstract

This paper investigates the computational relevance of the Energy-Conserving Sampling and Weighting (ECSW) method for hyper-reduction in reduced-order models (ROMs), with particular attention to modern Graphical Processing Unit (GPU)-accelerated and deep-learning-based frameworks. While hyper-reduction techniques such as ECSW are traditionally introduced to enable tractable evaluation of nonlinear terms in Galerkin ROMs, their actual benefit for computable nonlinearities with traditional Proper Orthogonal Decomposition (POD)–Galerkin ROMs combined with deep learning and on deep-learning GPU hardware remains a question. We provide a full ECSW implementation for finite element Navier–Stokes solvers, compatible with automatic differentiation and batched tensor operations, and derive exact operation counts for both standard tensorial and hyper-reduced formulations. Theoretical analysis shows that the computational complexity scales as

for tensorial ROMs and

O rm $ \mathcal{O}(rm) $ Orm

for ECSW, where

r $ r $ r

is the reduced dimension and

m $ m $ m

the number of retained quadrature points. Extensive CPU and GPU benchmarks are conducted on the fluidic pinball configuration to assess asymptotic scaling, memory effects, and hardware bottlenecks. Results confirm the linear complexity and efficiency of ECSW on CPUs but reveal that, on GPUs, the regular tensorial formulation can outperform hyper-reduction for common reduction dimensions due to highly optimized batched matrix kernels and limited bandwidth constraints. Overall, the study clarifies when hyper-reduction remains advantageous in practice and provides concrete guidelines for integrating ECSW within deep-learning-augmented ROM frameworks.