AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies

PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also selecting appropriate discretization schemes and solver configurations, and correctly implementing the resulting formulations in finite-element method (FEM) libraries. Existing code generation benchmarks mainly evaluate syntactic correctness, or success on predefined test cases. To our knowledge, there is currently no publicly available benchmark specifically for PDE-to-solver code generation, and general-purpose code benchmarks do not fully capture the unique challenges of numerical PDE solution, such as ensuring solver accuracy, efficiency, and compatibility with professional FEM libraries. We introduce PDEAgent-Bench, to the best of our knowledge, the first multi-metric, multi-library benchmark for PDE-to-solver code generation. PDEAgent-Bench contains 645 instances across 6 mathematical categories and 11 PDE families, with common FEM libraries for DOLFINx, Firedrake, and deal.II. Each instance provides an agent-facing problem specification, a reference solution on a prescribed evaluation grid, and case-specific accuracy and runtime targets. PDEAgent-Bench adopts a staged evaluation framework in which generated solvers must sequentially pass executability, numerical accuracy, and computational efficiency checks. Experiments with representative LLMs and code agents show that models can often produce runnable code, but their pass rate drops substantially once accuracy and efficiency requirements are enforced. These results indicate that current agents remain limited in producing numerically reliable and efficient PDE solvers, and that PDEAgent-Bench provides a reproducible testbed grounded in the practical requirements of numerical PDE solving.

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.

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation. RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.