LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation

Traditional symbolic regression methods for discovering partial differential equations (PDEs) often struggle with two major issues: the inability to incorporate domain-specific physical knowledge and the tendency of discrete numerical differentiation to amplify high-frequency noise. This paper addresses these bottlenecks by proposing LLM-PDESR, a framework that leverages the reasoning capabilities of Large Language Models (LLMs) alongside a mathematically rigorous, noise-robust evaluation pipeline.

The authors introduce a three-part framework:

1. LLM-Driven Hypothesis Generation: The LLM acts as an architect, proposing candidate symbolic skeletons for PDEs while using parameter placeholders for coefficients. The search space is constrained to ensure physical validity.
2. Robust Parameter Identification: To avoid the noise amplification inherent in finite difference methods, the framework uses $C^4$-continuous quintic splines for differentiation. It then employs a Subdomain Weighted Residual (SWR) evaluation, which acts as a natural low-pass filter by integrating residuals against smooth test functions over local domains.
3. Pareto-Driven Evaluation: The system uses a feedback loop that evaluates candidates based on both predictive accuracy (negative mean squared error) and structural parsimony (complexity of the abstract syntax tree). The top-performing candidates are fed back into the LLM to iteratively refine the search.

LLM-PDESR was evaluated on 23 canonical PDEs and five structurally novel equations designed to test genuine discovery rather than dataset memorization. The framework demonstrated superior structural recovery and noise resilience compared to state-of-the-art methods like SGA-PDE, DISCOVER, and EqGPT. Notably, the authors successfully extracted a 1D dynamical surrogate for atmospheric circulation directly from noisy ERA5 climate data, confirming the framework's ability to capture invariant physical mechanisms in real-world, high-noise environments.