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

2026-07-11

AI Summary

This work addresses the challenge of accurately discovering governing partial differential equations (PDEs) from noisy data, which is hindered by an expansive symbolic search space, lack of prior knowledge, and noise amplification during differentiation. The authors propose a novel framework that integrates large language models (LLMs) with rigorous mathematical evaluation: LLMs generate plausible equation structures, while C⁴-continuous quintic splines enable robust numerical differentiation. A subdomain-weighted residual formulation acts as a natural low-pass filter to suppress noise, and a Pareto-driven feedback mechanism balances predictive accuracy against model simplicity. Evaluated on 28 PDEs—including five with previously unreported structures—the method substantially outperforms existing approaches. It also successfully extracts an interpretable one-dimensional dynamical surrogate model from ERA5 reanalysis data, demonstrating exceptional capability in structural recovery, noise resilience, and generalization.

Abstract

Discovering governing partial differential equations (PDEs) from noisy observational data is a fundamental challenge in scientific machine learning. Traditional symbolic regression (SR) methods often struggle to identify accurate equations within vast combinatorial search spaces, largely due to their inability to incorporate essential domain-specific prior knowledge. Furthermore, reliance on pointwise evaluations and discrete finite differences inherently amplifies high-frequency noise, creating deceptive fitness landscapes that derail the optimization process. To resolve these bottlenecks, we propose LLM-PDESR, a framework that integrates the structural hypothesis generation of Large Language Models (LLMs) with a mathematically rigorous evaluation environment. By employing C^4-continuous quintic splines for robust differentiation and subdomain weighted residuals as natural low-pass filters, our approach effectively mitigates the fitness landscape distortion that plagues existing methods. A Pareto-driven feedback loop then enables the LLM to iteratively refine candidate equations, balancing predictive accuracy with structural parsimony. We evaluate LLM-PDESR on 23 canonical PDEs and five structurally novel equations (including a multivariate system) specifically designed to preclude dataset memorization and test true discovery capabilities. Demonstrating real-world applicability, the framework successfully extracts a consistent structural skeleton for an interpretable 1D dynamical surrogate (1D-CACE) directly from noisy ERA5 reanalysis data. Extensive experiments and out-of-distribution testing confirm that LLM-PDESR significantly outperforms state-of-the-art methodologies in structural recovery, noise resilience, and the avoidance of spurious complexity and equation bloat.

Problem

• PDE discovery
• symbolic regression
• noisy data
• fitness landscape distortion
• domain knowledge

Innovation

• LLM-guided symbolic regression
• subdomain weighted residuals
• C⁴-continuous splines
• Pareto-driven feedback
• robust PDE discovery