LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation
2026-07-12 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed LLM-PDESR, a new approach to find the underlying partial differential equations (PDEs) that govern physical systems from noisy data. Their method uses large language models to suggest possible equation structures and a special mathematical technique with smooth splines to handle noise better. By combining these with an iterative feedback process, they can improve both accuracy and simplicity of the discovered equations. They tested their method on many standard and new PDEs, showing it works better than previous methods, even on real-world weather data. This approach helps reveal meaningful mathematical models from messy observations without getting tricked by noise.
Partial Differential EquationsSymbolic RegressionLarge Language ModelsQuintic SplinesNoise FilteringFitness LandscapePareto OptimizationScientific Machine LearningERA5 Reanalysis Data
Authors
Jinyang Du, Hao Ma, Xiaohu Shi, Bo Yang, Yanchun Liang, Heow Pueh Lee, Chunguo Wu
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.