Leveraging Mathematical Reasoning of LLMs for Efficient GPU Thread Mapping

TL;DR

This paper demonstrates how Large Language Models can automate the derivation of efficient GPU thread mappings for complex geometries, significantly reducing energy and time costs in GPU computing.

Contribution

It introduces a novel method using LLMs for symbolic reasoning to automate GPU thread mapping derivations, outperforming traditional methods in accuracy and efficiency.

Findings

LLMs successfully infer exact mapping equations for complex domains.

One-time inference incurs high energy cost but enables massive savings during GPU execution.

Current models face limitations with highly recursive 3D fractals like the Menger Sponge.

Abstract

Mapping parallel threads onto non-box-shaped domains is a known challenge in GPU computing; efficient mapping prevents performance penalties from unnecessary resource allocation. Currently, achieving this requires significant analytical human effort to manually derive bespoke mapping functions for each geometry. This work introduces a novel approach leveraging the symbolic reasoning of Large Language Models (LLMs) to automate this derivation entirely through in-context learning. Focusing on state-of-the-art open-weights models, we conducted a rigorous comparative analysis across spatial domains of increasing complexity. Our results demonstrate that modern local LLMs successfully infer exact O(1) and O(log N) mapping equations for complex 2D/3D dense domains and 2D fractals, vastly outperforming traditional symbolic regression methods. Crucially, we profile the energetic viability of this approach on high-performance infrastructure, distinguishing between the code-generation and execution phases. While one-time inference incurs a high energy penalty -- particularly for reasoning-focused models like DeepSeek-R1 -- this is a single upfront investment. Once integrated, the generated analytical kernels eliminate block waste entirely, yielding massive energy and time savings (e.g., up to 4833x speedup and 2890x energy reduction) during actual GPU workloads. Finally, we identify a current "reasoning ceiling" when these models face highly recursive 3D fractals (e.g., the Menger Sponge). This limitation benchmarks the present maturity of open-weight architectures, charting a viable path toward fully automated, energy-efficient GPU resource optimization.