Symbolic regression is a machine learning technique that can learn the governing formulas of data and thus has the potential to transform scientific discovery. However, symbolic regression is still limited in the complexity and dimensionality of the systems that it can analyze. Deep learning on the other hand has transformed machine learning in its ability to analyze extremely complex and high-dimensional datasets. We propose a neural network architecture to extend symbolic regression to parametric systems where some coefficient may vary but the structure of the underlying governing equation remains constant. We demonstrate our method on various analytic expressions, ODEs, and PDEs with varying coefficients and show that it extrapolates well outside of the training domain. The neural network-based architecture can also integrate with other deep learning architectures so that it can analyze high-dimensional data while being trained end-to-end. To this end we integrate our architecture with convolutional neural networks to analyze 1D images of varying spring systems.
Discovering equations from data: symbolic regression in dynamical systems
Institute of Sciences Mathematics and Computation, Universidade de São Paulo-Campus de São Carlos（数学与计算科学研究院，圣保罗大学-萨o卡洛斯校区）；Statistic Department, State University of Maringá（统计系，马林加州立大学）；Department of Mathematics Applied and Statistics, Institute of Sciences Mathematics and Computation, Universidade de São Paulo-Campus de São Carlos（应用数学与统计系，数学与计算科学研究院，圣保罗大学-萨o卡洛斯校区）
Discovering Continuous-Time Memory-Based Symbolic Policies using Genetic Programming
Department of Machine Learning and Neural Computing（机器学习与神经计算系）；Donders Institute for Brain, Cognition and Behaviour（大脑、认知与行为研究所）；Radboud University（拉德堡德大学）
Neuro-Symbolic Imitation Learning: Discovering Symbolic Abstractions for Skill Learning
Intelligent Autonomous Systems, TU Darmstadt, Germany（图腾达姆施塔特大学智能自主系统研究所）；German Research Center for AI, Germany（德国人工智能研究中心）；Hessian Centre for Artificial Intelligence, Germany（黑森州人工智能中心）；Honda Research Institute EU, Germany（本田欧洲研究院）
Discovering physical laws with parallel symbolic enumeration
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学人工智能学院）；School of Engineering Science, University of Chinese Academy of Sciences（中国科学院大学工程科学学院）；State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences（中国科学院力学研究所非线性力学重点实验室）；Department of Computer Science and Engineering, HKUST（香港科技大学计算机科学与工程系）
Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)
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）
Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regression
School of Ocean and Civil Engineering, Shanghai Jiao Tong University（上海交通大学海洋与土木工程学院）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（东方理工大学宁波数字孪生研究院）
Physics Symbolic Learner for Discovering Ground-Motion Models Via NGA-West2 Database
Beijing University of Technology（北京工业大学）；Institute of Geophysics, China Earthquake Administration（中国地震局地球物理研究所）；Institute of Geomechanics, Chinese Academy of Geological Sciences（中国地质科学院地质力学研究所）
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）
Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation
Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea（韩国首尔 Chung-Ang 大学人工智能系）；Daejin University, Pocheon, Republic of Korea（韩国坡州市 Daejin 大学）
Discovering ordinary differential equations that govern time-series
Helmholtz AI（亥姆霍兹人工智能中心）；Technical University of Munich（慕尼黑工业大学）；Apple（苹果公司）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；DeepMind（深度思维）；OpenAI（开放人工智能）
Scalable Neural Symbolic Regression using Control Variables
School of Computer Science（计算机科学学院）；GaTech（佐治亚理工学院）；School of Computing and Data Science（计算与数据科学学院）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；William & Mary（威廉玛丽学院）；Department of EECS（电子工程与计算机科学系）；University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；Wake Forest University（维克森林大学）
Learning Parametric Nitrogen Fertilizer Response Curves Using Neuro Symbolic Regression
Aston Centre for Artificial Intelligence Research and Application（阿斯顿人工智能研究与应用中心）；Aston University（阿斯顿大学）；Gianforte School of Computing（吉安福特计算学院）；Montana State University（蒙大拿州立大学）
Discovering Mathematical Formulas from Data via GPT-guided Monte Carlo Tree Search
ANNLAB, Institute of Semiconductors, Chinese Academy of Sciences（中国科学院半导体研究所ANNLAB）；School of Integrated Circuits, University of Chinese Academy of Sciences（中国科学院大学集成电路学院）；School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences（中国科学院大学电子电气与通信工程学院）
Finite Expression Methods for Discovering Physical Laws from Data
Department of Statistics and Data Science（统计与数据科学系）；Yale University（耶鲁大学）；Department of Mathematics（数学系）；University of Florida（佛罗里达大学）；Department of Computer Science（计算机科学系）；University of Maryland（马里兰大学）；College Park（学院公园）
SymbolFit: Automatic Parametric Modeling with Symbolic Regression
University of Pennsylvania（宾夕法尼亚大学）；European Organization for Nuclear Research (CERN)（欧洲核子研究中心）；University of Cambridge（剑桥大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California San Diego（加州大学圣地亚哥分校）；Massachusetts Institute of Technology（麻省理工学院）；Institute for Artificial Intelligence（人工智能研究所）
AutoSINDy: Discovery of Nonlinear Dynamics with Automated Library Construction
Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA（数据科学与分析研究所，俄克拉荷马大学，诺曼，OK，USA）；School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA（工业与系统工程学院，俄克拉荷马大学，诺曼，OK，USA）
UniSymNet: A Unified Symbolic Network Guided by Transformer
School of Mathematical Sciences, East China Normal University（东华大学数学科学学院）；Institute of Artificial Intelligence, Beihang University（北航人工智能研究院）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究所）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学先进跨学科研究学院）
Celestial Machine Learning: Discovering the Planarity, Heliocentricity, and Orbital Equation of Mars with AI Feynman
National University of Singapore（新加坡国立大学）；Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局（A*STAR）新加坡制造技术研究院）
Symbolic regression for defect interactions in 2D materials
HSE University（高等经济大学）；Institute for Functional Intelligent Materials, National University of Singapore（新加坡国立大学功能智能材料研究所）；National University of Singapore（新加坡国立大学）；Constructor University（康斯特大学）
Learning Interpretable Network Dynamics via Universal Neural Symbolic Regression
College of Computer Science and Technology, Jilin University（吉林大学计算机科学与技术学院）；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）
College of Computer Science and Technology, Jilin University（吉林大学计算机科学与技术学院）；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）
