Leveraging Mathematical Reasoning of LLMs for Efficient GPU Thread Mapping

2026-04-11

AI Summary

This work addresses the inefficiency and severe resource waste in mapping GPU parallel threads onto non-rectangular computational domains. For the first time, it leverages the symbolic reasoning and in-context learning capabilities of locally deployed open-source large language models (LLMs) to automatically synthesize exact O(1) and O(log N) thread mapping functions, replacing traditional manual derivation. The proposed approach significantly outperforms existing symbolic regression techniques on complex 2D/3D and fractal domains, while also revealing current LLMs’ limitations in highly recursive 3D fractal reasoning. Experimental results demonstrate that the generated block-waste-free analytical kernel functions achieve up to a 4,833× speedup and a 2,890× reduction in energy consumption under real GPU workloads, confirming their breakthrough advantages in both performance and energy efficiency.

Technology Category

• Machine Learning: Kernel Methods
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Mapping parallel threads onto non-box-shaped domains is a known challenge in GPU computing that, if done efficiently, can prevent severe performance penalties from allocating unnecessary computational resources. Currently, achieving this optimal efficiency requires significant analytical human time and effort to manually derive bespoke mapping functions for each specific geometry. This work introduces a novel approach leveraging the symbolic reasoning capabilities of Large Language Models (LLMs) to automate this mathematical derivation process 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 which systematically failed at this discrete task. Crucially, we profile the energetic viability of this approach on high-performance infrastructure, drawing a clear distinction between the code-generation and execution phases. While the one-time inference of the equation 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 repeated 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 tested in this work (e.g., the Menger Sponge). This limitation establishes a clear benchmark for the maturity of open-weight architectures, charting a viable and sovereign path toward fully automated, energy-efficient GPU resource optimization.

Problem

• GPU thread mapping
• non-box-shaped domains
• mathematical reasoning
• Large Language Models
• symbolic regression

Innovation

• Large Language Models
• GPU thread mapping
• symbolic reasoning
• energy-efficient computing
• fractal domains