Agents must now identify the PDE themselves: RSI-PDEbench tests them on 1,001 cases, lifting mean pass rates by up to 181.9% across four models
Synopsis
Recovering a partial differential equation from sampled solution data demands expert work in equation discovery, coefficient estimation and numerical solver configuration. RSI-PDEbench turns that workflow into a benchmark of 1,001 cases over 42 equation templates, split into noise-free, 50% Gaussian-noise and unseen-template-with-shifted-coefficient settings; the accompanying RSI-PDE agent shortlists candidate equations by residual, refines coefficients and solvers with relative feedback, and keeps improvements behind a validation gate. Across four frontier models its mean pass rate rises 106.5–181.9% over single-run baselines and 3.0–19.6% over native agent harnesses.
Interpretation
PDE-solving agents now face a harder task: neither the equation family nor its coefficients is given, only a finite equation library, 256 space-time samples and known forcing, initial and boundary functions, and the agent must identify the equation and return an executable numerical solution. Earlier PDE agent benchmarks mostly generate solvers for specified equations; for example, PDEAgent-Bench provides 645 solver-generation instances for specified PDEs, and CodePDE studies solver synthesis for specified equations. RSI-PDEbench releases 1,001 PDE and fractional-PDE cases spanning 42 templates, which the authors describe as the largest joint PDE/FPDE benchmark for coding agents tasked with equation-family identification, parameter estimation and solver construction.
RSI-PDE organizes exploration into a loop: inside a sandbox the agent substitutes a reconstructed or computed field into candidate equations and fits coefficients, keeps up to five low-residual families, then uses forward solves and public relative-error feedback to separate hypotheses and repair coefficients and discretizations. Residuals only narrow the family search; the final judgment comes from relative-error tests, and hidden references never guide online equation choice. The validation gate retains changes to the solver, instruction policy and memory only when, among other conditions, macro relative error drops by more than 0.5% and no class-average error rises above 5%, while the foundation model and executable supervisor stay fixed.
Across four frontier models, RSI-PDE's mean pass rate exceeds both single-run and native-harness baselines in all three settings, while recorded model calls across the 12 model-setting pairs fall from 3,547 to 1,884. Mean pass rates for single run/native/RSI-PDE are 28.7/78.6/81.0% on Full, 26.5/53.3/63.8% on Noisy and 37.2/72.9/76.9% on Hard-OOD. Absolute gains over native harnesses are 2.4, 10.5 and 4.0 percentage points; calls are lower in 8 of 12 pairs, while on Noisy Opus 5.5, GPT-6.1 and Sonnet 5.5 use more.
Perspective
The demonstrated scope is 1,001 synthetic PDE/FPDE cases over 42 templates in one- and two-dimensional domains, compared across four frontier models under the Full, Noisy and Hard-OOD settings; the authors call for repeated runs, residual-specific ablations and fresh hidden evaluations to clarify which parts of the workflow generalize most reliably.
Most model-setting comparisons have one run, and on Full the native Opus 5.5 three-run range of 187–198 passes is larger than its three-case gap to RSI-PDE; residual guidance has no isolated ablation, and a low residual does not guarantee the correct law survives the shortlist; Noisy cases come from a regenerable public release rather than a fresh secret holdout, and Hard-OOD is feasibility-filtered; some dollar costs are retrospective stop-rule replay estimates, not accuracy and cost measured together in a fresh stopped run.