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In drug discovery, molecular optimization aims to iteratively refine a lead compound to improve molecular properties while preserving structural similarity to the original molecule. However, each oracle evaluation is expensive, making sample efficiency a key challenge for existing methods under a limited oracle budget. Trial-and-error approaches require many oracle calls, while methods that leverage external knowledge tend to reuse familiar templates and struggle on challenging objectives. A key missing piece is long-term memory that can ground decisions and provide reusable insights for future optimizations. To address this, we present MolMem (\textbf{Mol}ecular optimization with \textbf{Mem}ory), a multi-turn agentic reinforcement learning (RL) framework with a dual-memory system. Specifically, MolMem uses Static Exemplar Memory to retrieve relevant exemplars for cold-start grounding, and Evolving Skill Memory to distill successful trajectories into reusable strategies. Built on this memory-augmented formulation, we train the policy with dense step-wise rewards, turning costly rollouts into long-term knowledge that improves future optimization. Extensive experiments show that MolMem achieves 90\% success on single-property tasks (1.5$\times$ over the best baseline) and 52\% on multi-property tasks using only 500 oracle calls. Our code is available at this https URL.