Identifying and solving partial differential equations (PDEs) from sampled solution data often requires substantial expert effort in equation discovery, coefficient estimation, and numerical solver configuration, especially under noise and distribution shifts. Can large language model (LLM) agents automate this workflow accurately and efficiently? To investigate this question, we introduce \bench{}, a benchmark of 1,001 cases spanning 42 equation templates, including both classical and fractional PDEs. It provides three evaluation settings: Full evaluates performance on noise-free solution data; Noisy tests robustness to Gaussian noise with an RMS equal to 50\% of the clean-signal RMS; and Hard-OOD tests challenging out-of-distribution generalization to unseen equation templates and shifted coefficient ranges. We further introduce RSI-PDE, a recursively self-improving agent harness that shortlists candidate equations by residual loss, uses relative $L^2$ feedback to refine coefficients and numerical solvers, and applies a validation gate to policy and memory updates. We compare RSI-PDE with single-run baselines and native agent harnesses (Claude Code or Codex) across four frontier models. Across the three settings, RSI-PDE improves the pass rate averaged across the four models by 106.5--181.9\% relative to single-run baselines and by 3.0--19.6\% relative to native harnesses, while reducing total recorded model calls by 46.9\% across the 12 model--setting pairs. These results highlight the potential of validated recursive self-improvement in LLM agents to advance scientific computing.

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