PDEDiscoveryBench: Evaluating Differential Equation Discovery from Spatiotemporal Data

Discovering a partial differential equation (PDE) from observed fields requires choosing which terms may enter the law. Existing methods address this task, but published results rarely report how much prior information a method received, and to our knowledge no benchmark controls that information across method types. We introduce PDEDiscoveryBench, 337 ODE and PDE systems in one machine-readable format: 332 with reference equations and five unlabeled blood-flow fields simulated with computational fluid dynamics (CFD). Each labeled system is posed at three prior levels, from generic assumptions (L1) to equation-informed guidance (L3), written as natural-language prior cards that each method receives in its native form, and at four noise tiers. We evaluate seven sparse-regression, symbolic-search and large language model (LLM)-based methods by exact structural recovery and held-out normalized mean squared error (NMSE). Exact recovery is higher at L3 than at L1 for every method at every noise tier. Four case studies examine the prior from different sides: the data did not override a wrong prior card; LLM-PDESR recovered a KdV equation exactly only under its published, more specific specification; telling two LLM agents the noise model raised their exact recovery; and LLM agents given no physical description returned relations containing the Stokes balance for all five CFD fields. PDEDiscoveryBench makes prior information an explicit, controlled variable, so that new classical and LLM-based methods can be compared under the same priors and noise; we release its data, prior cards and scoring code. Our results suggest that discovery claims should state the prior behind them.