Domain-Specific Language Boosts Neural PDE Solver Auto-Design.

Automating the design of neural PDE (Partial Differential Equation) solvers is challenging because the search space of unrestricted code is vast and mostly invalid. Large Language Models (LLMs) waste significant capacity navigating syntactic errors, semantic incompatibilities, or numerical instabilities when directly generating code. To overcome this, ADSL-PDE introduces a structured search state that bridges the gap between high-level solver concepts and executable code. ADSL-PDE represents the critical functional decisions for a neural PDE solver, such as architecture, physical constraints, objectives, sampling, and optimization, while abstracting away intricate implementation details. A deterministic compiler then maps each valid search state to a functional solver. This effectively reshapes the search space, eliminating invalid programs and increasing the density of meaningful candidates, thereby preserving the compositional freedom needed for novel designs. Built upon this refined representation, an evolutionary agent iteratively proposes, evaluates, and refines solver search states using empirical feedback. This method, operating over design decisions rather than raw code, significantly improves search efficiency and optimization stability across various PDE benchmarks, achieving over 52% improvement within the first ten iterations. This highlights the importance of effective search representations for LLM-driven auto-design.