From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs
Department of Computer Science, University of Maryland, College Park, Maryland, USA（计算机科学系，马里兰大学， College Park, Maryland, USA）；Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA（数学系，田纳西大学， Knoxville, Tennessee, USA）；Department of Mathematics, and Department of Computer Science, University of Maryland, College Park, Maryland, USA（数学系和计算机科学系，马里兰大学， College Park, Maryland, USA）
Rohan Bhatnagar, Ling Liang, Krish Patel, Haizhao Yang
受人工智能（AI）在众多领域取得显著成功的推动，将 AI 应用于求解通常表述为偏微分方程（PDE）的科学问题，正受到越来越多的关注。现有研究大多集中在解的理论性质（如适定性、正则性和连续性）以及由 AI 驱动的直接 PDE 求解方法上，而揭示这些方程内部符号关系的挑战在很大程度上仍未得到探索。本文提出利用大语言模型（LLM）学习此类符号关系。结果表明，LLM 能够在理论和数值两方面利用 PDE 中的符号信息，有效预测 PDE 求解所涉及的算子。此外，本文还表明，发现这些符号关系可以显著提升用于寻找 PDE 解解析近似的符号机器学习方法的效率和准确性，并提供一个完全可解释的求解流程。本工作为理解科学问题的符号结构并推进其求解过程开辟了新途径。
英文摘要：
Motivated by the remarkable success of artificial intelligence (AI) across diverse fields, the application of AI to solve scientific problems, often formulated as partial differential equations (PDEs), has garnered increasing attention. While most existing research concentrates on theoretical properties (such as well-posedness, regularity, and continuity) of the solutions, alongside direct AI-driven methods for solving PDEs, the challenge of uncovering symbolic relationships within these equations remains largely unexplored. In this paper, we propose leveraging large language models (LLMs) to learn such symbolic relationships. Our results demonstrate that LLMs can effectively predict the operators involved in PDE solutions by utilizing the symbolic information in the PDEs both theoretically and numerically. Furthermore, we show that discovering these symbolic relationships can substantially improve both the efficiency and accuracy of symbolic machine learning for finding analytical approximation of PDE solutions, delivering a fully interpretable solution pipeline. This work opens new avenues for understanding the symbolic structure of scientific problems and advancing their solution processes.
Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation
School of Mathematical Sciences, East China Normal University（东华大学数学科学学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Institute of Artificial Intelligence, Beihang University（北航人工智能研究所）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究所）；Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学先进交叉学科研究院）
LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation
College of Computer Science and Technology, Jilin University（吉林大学计算机科学与技术学院）；School of Big Data and Artificial Intelligence, Guangdong University of Finance and Economics（广东财经大学大数据与人工智能学院）；Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University（吉林大学符号计算与知识工程教育部重点实验室）；School of Computer Science, Zhuhai College of Science and Technology（珠海科技学院计算机科学学院）；Department of Mechanical Engineering, National University of Singapore（新加坡国立大学机械工程系）
Peking University（北京大学）；Eastern Institute for Advanced Study（东方高等研究院）；Eastern Institute of Technology（东方理工大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Southern University of Science and Technology（南方科技大学）
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a structure-aware recurrent neural network agent is designed and seaml…
Physics-informed Neural Networks (PINNs) have been widely used to obtain accurate neural surrogates for a system of Partial Differential Equations (PDE). One of the major limitations of PINNs is that the neural solutions are challenging to interpret, and are often treated as black-box solvers. While Symbolic Regression (SR) has been studied extensively, very few works exist which generate analytical expressions to directly perform SR for a system of PDEs. In this work, we introduce an end-to-end framework for obtaining mathematical expressions for solutions of PDEs. We use a trained PINN to ge…
Peng Cheng Laboratory（鹏城实验室）；University of Washington（华盛顿大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）
Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which are essential for deepening our understanding of physical processes and predicting future responses. However, the PDEs of many real-world problems are uncertain, which calls for PDE discovery. We propose the symbolic genetic algorithm (SGA-PDE) to discover open-form PDEs directly from data without prior knowledge about the equation structure. SGA-PDE focuses on the representation and optimization of PDE. Firstly, SGA-PDE uses symbolic mathematics to realize the flexible representatio…
SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校数学系）；Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校机械与航空航天工程系）；Office of Advanced Research Computing (OARC), University of California, Los Angeles, Los Angeles, CA, USA（加州大学洛杉矶分校高级计算研究办公室）
Robust identifiability for symbolic recovery of differential equations
Ludwig-Maximilians-Universität München（慕尼黑大学）；University of Tromsø（特罗姆瑟大学）；DLR-German Aerospace Center（德国航空航天中心）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being advanced by machine learning, including: 1) the discovery of new governing PDEs and coarse-grained approximations for complex natural and engineered systems, 2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and 3) representing sol…
Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains
Department of Mathematics, Purdue University（普渡大学数学系）；Department of Mathematics and Department of Computer Science, University of Maryland College Park（马里兰大学学院市数学系和计算机科学系）
Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data
School of Software Engineering, Chengdu University of Information Technology（成都信息科技学院软件工程学院）；School of Statistics, Chengdu University of Information Technology（成都信息科技学院统计学院）；College of Applied Mathematics, Chengdu University of Information Technology（成都信息科技学院应用数学学院）；Key Laboratory of Mathematical Meteorology, Chengdu University of Information Technology（成都信息科技学院数学气象重点实验室）
Neuro-Symbolic AI for Analytical Solutions of Differential Equations
Seminar for Applied Mathematics, ETH Zurich, Switzerland（应用数学研讨会，苏黎世联邦理工学院，瑞士）；ETH AI Center, Zurich, Switzerland（苏黎世联邦理工学院人工智能中心，瑞士）；IBM Research Europe, Zurich, Switzerland（IBM欧洲研究院，苏黎世，瑞士）；Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland（大气与气候科学研究所，苏黎世联邦理工学院，瑞士）；Swiss Data Science Center, ETH Zurich, Switzerland（瑞士数据科学中心，苏黎世联邦理工学院，瑞士）
Physics-constrained robust learning of open-form partial differential equations from limited and noisy data
College of Engineering, Peking University（北京大学工学院）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（东方理工大学宁波数字孪生研究院）；School of Environmental Science and Engineering, Southern University of Science and Technology（南方科技大学环境科学与工程学院）；Eastern Institute for Advanced Study, Eastern Institute of Technology（东方理工大学东方高等研究院）
PI-MFM: Physics-informed multimodal foundation model for solving partial differential equations
Department of Statistics and Data Science, Yale University（统计与数据科学系，耶鲁大学）；Department of Applied Mathematics and Statistics, Johns Hopkins University（应用数学与统计学系，约翰霍普金斯大学）；Department of Applied Computational Mathematics and Statistics, University of Notre Dame（应用计算数学与统计学系，圣母大学）；Department of Mathematics, University of California Los Angeles（数学系，加州大学洛杉矶分校）；Department of Chemical and Environmental Engineering, Yale University（化学与环境工程系，耶鲁大学）
