Improving Auto-Design of Neural PDE Solvers with a Domain-Specific Language

Numerical solvers for partial differential equations (PDEs) are core computational tools in science and engineering. Building reliable PDE solvers requires not only executable code, but a numerical solver strategy, a set of decisions about discretization, stabilization, solver configuration, and resolution control, that matches the PDE structure. Recent LLM-based coding agents have begun to reduce the programming burden by generating and debugging solver implementations. However, they typically move directly from a PDE problem to solver code, leaving the solver strategy implicit in implementation details. Feedback from a failed solve is therefore routed back to code edits rather than to the underlying strategy, so numerical decisions remain hard to check before code is generated and hard to revise using numerical evidence when it fails. To address this limitation, we propose AutoPDE, a code agent that maintains the solver strategy as an explicitly represented object throughout the solving process: an independent, inspectable object that is built before any code is written and can be revised, using numerical evidence, whenever a solve fails. AutoPDE builds and maintains this object in three stages, all drawing from a library of reusable PDE-solving skills: PDE analysis identifies the equation type and algebraic structure; numerical method selection chooses a numerical method that matches the analysis result and commits to a discretization, stabilization, and linear solver accordingly; and adaptive tuning runs low-cost pilot solves to calibrate resolution and tolerances under the prescribed accuracy and runtime budget. We evaluate AutoPDE on the PDE Agent Bench, where experimental results show that AutoPDE achieves a pass rate of $54.5$, improving over the strongest baseline by $14.2$ percentage points.

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.

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.