Accurately modeling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of complex ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and the Belousov Zhabotinsky (BZ) reaction.
An Introduction to Sparse Identification of Nonlinear Dynamics for Engineering Applications
Department of Aeronautics, Imperial College London（伦敦帝国理工学院航空系）；Department of Mechanical Engineering, University of Washington（华盛顿大学机械工程系）；NSF AI Institute in Dynamic Systems, University of Washington（华盛顿大学动态系统领域美国国家科学基金会人工智能研究所）
The SINDy algorithm has been successfully used to identify the governing equations of dynamical systems from time series data. In this paper, we argue that this makes SINDy a potentially useful tool for causal discovery and that existing tools for causal discovery can be used to dramatically improve the performance of SINDy as tool for robust sparse modeling and system identification. We then demonstrate empirically that augmenting the SINDy algorithm with tools from causal discovery can provides engineers with a tool for learning causally robust governing equations.
SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks
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，美国。）；Analytics Institute, University of Oklahoma, Norman, OK, USA. School of Computer Science, University of Oklahoma, Norman, OK, USA.（分析研究所，俄克拉荷马大学，诺曼，OK，美国。计算机科学学院，俄克拉荷马大学，诺曼，OK，美国。）
Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data (Brunton et al., PNAS, '16; Rudy et al., Sci. Adv. '17). Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in (arXiv:2005.04339) to the setting of partial differential equations (PDEs). The elimination of pointwise derivative approximations via the weak form enables effective machine-p…
Generalizing the SINDy approach with nested neural networks
Conservatoire National des Arts et Métiers, Laboratoire M2N, 75003, Paris；Université de Strasbourg, CNRS, Inria, IRMA, F-67000 Strasbourg, France；PRC, INRAE, CNRS, Université de Tours, Nouzilly, France；Aix--Marseille University, Aix-en-Provence, France；Universit\'e de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France
University of Maryland（马里兰大学）；University of Washington（华盛顿大学）；Imperial College（帝国理工学院）
Sparse system identification is the data-driven process of obtaining parsimonious differential equations that describe the evolution of a dynamical system, balancing model complexity and accuracy. There has been rapid innovation in system identification across scientific domains, but there remains a gap in the literature for large-scale methodological comparisons that are evaluated on a variety of dynamical systems. In this work, we systematically benchmark sparse regression variants by utilizing the dysts standardized database of chaotic systems. In particular, we demonstrate how this open-so…
Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for $\ell_1$-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian sp…
Data-driven system identification using quadratic embeddings of nonlinear dynamics
School of Mathematical & Computer Sciences, Heriot–Watt University（数学与计算机科学学院，赫里奥特- Watt大学）；Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University（数学科学麦克斯韦研究所，爱丁堡大学和赫里奥特- Watt大学）
From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification
AI Institute in Dynamic Systems, Department of Mechanical Engineering, University of Washington（动态系统人工智能研究所，机械工程系，华盛顿大学）；Department of Mechanical Engineering, University of Hawai‘i at Mānoa（机械工程系，夏威夷大学马诺阿分校）；Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista “Júlio de Mesquita Filho”（数学与计算系，科技学院，保罗斯州立大学“朱利奥·梅斯奎塔·弗洛雷斯”分校）
Department of Mechanical Engineering, Universidad de Chile, Beauchef 851, Santiago, Chile（智利大学机械工程系）；Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom（伦敦帝国理工学院航空系）；Department of Mechanical and Metallurgical Engineering, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile（智利天主教大学机械与冶金工程系）；Department of Hydraulic and Environmental Engineering, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile（智利天主教大学水利与环境工程系）
University of Washington（华盛顿大学）；Arts et Métiers Institute of Technology（国立高等工程技术学院）
Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differ…
Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics
Department of Mechanical and Aeronautical Engineering, University of Pretoria（机械与航空工程系，普里蒂亚大学）；School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand（机械、工业与航空工程学院，威特沃特斯兰大学）
Mesh-free sparse identification of nonlinear dynamics
Paul G. Allen School of Computer Science & Engineering University of Washington（保罗·G·艾伦计算机科学与工程学院华盛顿大学）；Applied Mathematics and Electrical and Computer Engineering University of Washington（应用数学与电气与计算机工程华盛顿大学）；Faculty of Mechanical Engineering Delft University of Technology（机械工程学院代尔夫特理工大学）
This work proposes an iterative sparse-regularized regression method to recover governing equations of nonlinear dynamical systems from noisy state measurements. The method is inspired by the Sparse Identification of Nonlinear Dynamics (SINDy) approach of {\it [Brunton et al., PNAS, 113 (15) (2016) 3932-3937]}, which relies on two main assumptions: the state variables are known {\it a priori} and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. The aim of this work is to improve the accuracy and robustness of SINDy in the prese…
Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China（非线性力学国家重点实验室，力学研究所，中国科学院，北京100190，中国）；School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China（中国科学院大学工程科学学院，北京100049，中国）；School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China（西北工业大学数学与统计学院，西安710072，中国）
Arizona State University（亚利桑那州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Discovery of dynamical systems from data forms the foundation for data-driven modeling and recently, structure-preserving geometric perspectives have been shown to provide improved forecasting, stability, and physical realizability guarantees. We present here a unification of the Sparse Identification of Nonlinear Dynamics (SINDy) formalism with neural ordinary differential equations. The resulting framework allows learning of both "black-box" dynamics and learning of structure preserving bracket formalisms for both reversible and irreversible dynamics. We present a suite of benchmarks demonst…
LeARN: Learnable and Adaptive Representations for Nonlinear Dynamics in System Identification
Visual Computing Lab, Indian Institute of Science（印度科学院视觉计算实验室）；Department of Electrical and Electronics Engineering, BITS Pilani Hyderabad Campus（BITS Pilani Hyderabad校区电子与电气工程系）
Interpretable neural network system identification method for two families of second-order systems based on characteristic curves
Instituto de Física Rosario (IFIR), CONICET-UNR（罗萨里奥物理研究所（IFIR），CONICET-UNR）；Facultad de Ciencias Exactas, Ingeniería y Agrimensura (UNR)（精确科学、工程与土地测量学院（UNR））
Discovering governing equations of complex dynamical systems directly from data is a central problem in scientific machine learning. In recent years, the sparse identification of nonlinear dynamics (SINDy) framework, powered by heuristic sparse regression methods, has become a dominant tool for learning parsimonious models. We propose an exact formulation of the SINDy problem using mixed-integer optimization (MIO) to solve the sparsity constrained regression problem to provable optimality in seconds. On a large number of canonical ordinary and partial differential equations, we illustrate the …
On-line learning of dynamic systems: sparse regression meets Kalman filtering
Department of Information Engineering, University of Padova（信息工程系，帕多瓦大学）；Department of Neurology, University of Texas（神经病学系，德克萨斯大学）；Department of Applied Mathematics, University of Washington（应用数学系，华盛顿大学）
Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309 United States（应用数学系，科罗拉多大学，博尔德，CO 80309 美国）；Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 United States（洛斯阿拉莫斯国家实验室理论部，洛斯阿拉莫斯，NM 87545 美国）
The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work, we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al. for simultaneously (i) denoising the data, (ii) learning and parametrizing the noise probability distribution, and (iii) iden…
Discovery of Nonlinear Dynamics with Automated Basis Function Generation
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）
Discovering the dynamics of \emph{Sargassum} rafts' centers of mass
Department of Atmospheric Sciences, Rosenstiel School of Marine, Atmospheric \& Earth Science, University of Miami, Miami, Florida, USA.；Morgridge Institute for Research, University of Wisconsin, Madison, WI, USA.
We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system x' = f(x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-comp…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Stanford University（斯坦福大学）
We consider the problem of partial identification, the estimation of bounds on the treatment effects from observational data. Although studied using discrete treatment variables or in specific causal graphs (e.g., instrumental variables), partial identification has been recently explored using tools from deep generative modeling. We propose a new method for partial identification of average treatment effects(ATEs) in general causal graphs using implicit generative models comprising continuous and discrete random variables. Since ATE with continuous treatment is generally non-regular, we levera…
Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems
School of Computer and Mathematical Sciences, The University of Adelaide（计算机与数学科学学院，阿德莱德大学）；Australian Institute for Machine Learning, The University of Adelaide（澳大利亚机器学习研究所，阿德莱德大学）；Department of Biomedical Engineering, Northwestern University（生物医学工程系，西北大学）
How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit
Department of Mechanical Engineering, University of Washington（华盛顿大学机械工程系）；NSF AI Institute in Dynamic Systems, University of Washington（华盛顿大学NSF动态系统人工智能研究所）；Department of Aeronautics, Imperial College London（伦敦帝国理工学院航空系）
In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming the state variables are known a priori and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters whe…
