Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data

Data-driven discovery of partial differential equations (PDEs) offers a promising paradigm for uncovering governing physical laws from observational data. However, in practical scenarios, measurements are often contaminated by noise and limited by sparse sampling, which poses significant challenges to existing approaches based on numerical differentiation or integral formulations. In this work, we propose a Symbolic Graph Network (SGN) framework for PDE discovery under noisy and sparse conditions. Instead of relying on local differential approximations, SGN leverages graph message passing to model spatial interactions, providing a non-local representation that is less sensitive to high frequency noise. Based on this representation, the learned latent features are further processed by a symbolic regression module to extract interpretable mathematical expressions. We evaluate the proposed method on several benchmark systems, including the wave equation, convection-diffusion equation, and incompressible Navier-Stokes equations. Experimental results show that SGN can recover meaningful governing relations or solution forms under varying noise levels, and demonstrates improved robustness compared to baseline methods in sparse and noisy settings. These results suggest that combining graph-based representations with symbolic regression provides a viable direction for robust data-driven discovery of physical laws from imperfect observations. The code is available at https://github.com/CXY0112/SGN

KG-Augmented Executable CoT for Mathematical Coding

In recent years, large language models (LLMs) have excelled in natural language processing tasks but face significant challenges in complex reasoning tasks such as mathematical reasoning and code generation. To address these limitations, we propose KG-Augmented Executable Chain-of-Thought (KGA-ECoT), a novel framework that enhances code generation through knowledge graphs and improves mathematical reasoning via executable code. KGA-ECoT decomposes problems into a Structured Task Graph, leverages efficient GraphRAG for precise knowledge retrieval from mathematical libraries, and generates verifiable code to ensure computational accuracy. Evaluations on multiple mathematical reasoning benchmarks demonstrate that KGA-ECoT significantly outperforms existing prompting methods, achieving absolute accuracy improvements ranging from several to over ten percentage points. Further analysis confirms the critical roles of GraphRAG in enhancing code quality and external code execution in ensuring precision. These findings collectively establish KGA-ECoT as a robust and highly generalizable framework for complex mathematical reasoning tasks.