Neural PDE solver auto-design is fundamentally a search-space representation problem. In the space of unrestricted Python programs, valid solvers form an extremely sparse subset: most candidate programs are syntactically incorrect, semantically incompatible, or numerically unstable. Direct code generation therefore forces an LLM to spend most of its search capacity navigating implementation failures rather than reasoning about solver quality. ADSL-PDE addresses this challenge by introducing a structured search state between solver concepts and executable code. It represents the functional decisions that determine a neural PDE solver (architecture, physical constraints, objectives, sampling, and optimization) while abstracting away low-level implementation details. A deterministic compiler maps each valid search state to an executable solver. In effect, ADSL-PDE reshapes the search space: it removes large regions of invalid programs, increases the density of meaningful candidates, and preserves the compositional freedom needed to discover previously unseen designs. Solver evolution can thus operate over design decisions rather than code artifacts. Built on this representation, our evolutionary agent iteratively proposes, evaluates, and refines solver search states using empirical feedback. Across multiple PDE benchmarks, ADSL-PDE improves both search efficiency and optimization stability, achieving an improvement of more than 52% within the first ten evolution iterations. These results suggest a broader principle for LLM-driven auto-design: effective agents do not merely require stronger reasoning, but rather a search representation that concentrates exploration on valid and consequential decisions.
Neural architecture search (NAS) methods have grown increasingly efficient, yet they remain bounded by manually engineered search spaces that require substantial domain expertise and must be rebuilt for every new task. Large language models (LLMs) can generate architectures in an open-ended space, but how to optimally divide the labor between LLM-driven design and NAS-driven search remains unexplored. We propose a mechanism that bridges these two paradigms: an LLM produces a high-quality seed architecture, then decomposes it into a "slotted architecture", a scaffold with named, interchangeable module slots that automatically defines a bounded, task-specific search space for conventional NAS to explore, without manual engineering. We instantiate this mechanism in AgentNAS, a modular three-phase pipeline in which each component's contribution can be measured independently. On 17 tasks spanning classification, dense regression, segmentation, and multi-label tagging across diverse modalities (NAS-Bench-360 and Unseen NAS), AgentNAS establishes a new state of the art on 11 tasks, outperforming published baselines including task-specific expert designs. Ablation studies show that the two search mechanisms are broadly complementary: the LLM-generated seed already surpasses published baselines on the majority of tasks, and NAS delivers additional gains in most cases through combinatorial recombination across slots, a mode of search that independent LLM samples cannot replicate. These patterns hold across three LLMs of different capability levels, confirming that the division of labor is robust. Our code is available at https://github.com/alroimfebruary/AgentNAS.