Symbolic Recovery of Differential Equations: The Identifiability Problem
Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Technische Universität München（慕尼黑技术大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Munich Quantum Valley (MQV)（慕尼黑量子谷）；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）
Philipp Scholl, Aras Bacho, Holger Boche, Gitta Kutyniok
Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.
Efficient Symbolic Computations for Identifying Causal Effects
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学研究院）；Binghamton University（宾夕法尼亚州立大学布林茅尔分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
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）
Towards Identifiability of Interventional Stochastic Differential Equations
New York Genome Center（纽约基因组中心）；Department of Computer Science Columbia University（哥伦比亚大学计算机科学系）；Department of Applied Mathematics and Applied Physics Columbia University（哥伦比亚大学应用数学与应用物理系）；Institute of Computational Biology Helmholtz Munich（海德堡医学院计算生物学研究所）；Department of Mathematics Technische Universität München（慕尼黑技术大学数学系）；Department of Biomedical Engineering Columbia University（哥伦比亚大学生物医学工程系）；Irving Institute for Cancer Dynamics, Columbia University（哥伦比亚大学癌症动力学伊文斯研究所）
Identifying through Flows for Recovering Latent Representations
Institute of Data Science（数据科学研究所）；NUS Graduate School for Integrative Sciences and Engineering（新加坡国立大学综合科学与工程研究生院）；National University of Singapore（新加坡国立大学）
Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots
VIB, Center for Molecular Neurology（VIB分子神经学中心）；VIB, Center for AI and Computational Biology（VIB人工智能与计算生物学中心）；University of Antwerp（安特卫普大学）；University of Cambridge（剑桥大学）
DeSyR: A decoupled symbolic recovery framework for PDEs with PINN-guided structure discovery and physics-based coefficient refinement
Chengdu University of Information Technology（成都信息工程大学）；Sichuan University（四川大学）；Hong Kong Polytechnic University（香港理工大学）；Southwest Petroleum University（西南石油大学）
SRLR: Symbolic Regression based Logic Recovery to Counter Programmable Logic Controller Attacks
Beijing University of Posts and Telecommunications, China（北京邮电大学）；Aalborg University, Denmark（奥胡斯大学）；Singapore University of Technology and Design, Singapore（新加坡科技设计大学）
SymbolicArena: A Unified Infrastructure for Benchmark Distillation and Dynamic Evaluation in Symbolic Regression
School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；CDL, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所CDL）；Nanjing University of Science and Technology（南京理工大学）；School of Future Technology, University of Chinese Academy of Sciences（中国科学院大学未来技术学院）
FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression
School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China（浙江大学数学科学学院，杭州，浙江，中国）；Faculty of Medicine and Health Technology, Tampere University, Tampere, Finland（塔尔库大学医学与健康技术学院，塔尔库，芬兰）
The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery
University of Trieste（的里雅斯特大学）；National Institute of Oceanography and Applied Geophysics - OGS（国家海洋学与应用地球物理研究所 - OGS）；National Biodiversity Future Center（国家生物多样性未来中心）
Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子学与生物光子学重点实验室，物理学院，长春理工大学）
