GI-Agent integrates Large Language Models (LLMs) into Genetic Improvement (GI) to autonomously optimise computer program source code. GI-Agent uses an LLM to allow multi-generational evolutionary learning. Guided by a memory of past actions, known in AI as reflections, it exploits them to give insight and rationale behind software edits, giving better context-aware, less stochastic, mutations and crossovers. Reflections enable GI-Agent to reason about both earlier compile time and runtime successes and failures, and so refine its strategy over time. Integrated into the Magpie GI framework and evaluated on the SAT4J (Java) and MiniSAT (C++) benchmarks, GI-Agent consistently generates more viable and better variants. By combining few-shot prompting with structured search, GI-Agent demonstrates how LLMs can enhance automated program optimisation.