The sparse identification of nonlinear dynamical systems (SINDy) is a data-driven technique employed for uncovering and representing the fundamental dynamics of intricate systems based on observational data. However, a primary obstacle in the discovery of models for nonlinear partial differential equations (PDEs) lies in addressing the challenges posed by the curse of dimensionality and large datasets. Consequently, the strategic selection of the most informative samples within a given dataset plays a crucial role in reducing computational costs and enhancing the effectiveness of SINDy-based algorithms. To this aim, we employ a greedy sampling approach to the snapshot matrix of a PDE to obtain its valuable samples, which are suitable to train a deep neural network (DNN) in a SINDy framework. SINDy based algorithms often consist of a data collection unit, constructing a dictionary of basis functions, computing the time derivative, and solving a sparse identification problem which ends to regularised least squares minimization. In this paper, we extend the results of a SINDy based deep learning model discovery (DeePyMoD) approach by integrating greedy sampling technique in its data collection unit and new sparsity promoting algorithms in the least squares minimization unit. In this regard we introduce the greedy sampling neural network in sparse identification of nonlinear partial differential equations (GN-SINDy) which blends a greedy sampling method, the DNN, and the SINDy algorithm. In the implementation phase, to show the effectiveness of GN-SINDy, we compare its results with DeePyMoD by using a Python package that is prepared for this purpose on numerous PDE discovery
Sparse model identification enables the discovery of nonlinear dynamical systems purely from data; however, this approach is sensitive to noise, especially in the low-data limit. In this work, we leverage the statistical approach of bootstrap aggregating (bagging) to robustify the sparse identification of nonlinear dynamics (SINDy) algorithm. First, an ensemble of SINDy models is identified from subsets of limited and noisy data. The aggregate model statistics are then used to produce inclusion probabilities of the candidate functions, which enables uncertainty quantification and probabilistic…
We extend the data-driven method of Sparse Identification of Nonlinear Dynamics (SINDy) developed by Brunton et al, Proc. Natl. Acad. Sci USA 113 (2016) to the case of delay differential equations (DDEs). This is achieved in a bilevel optimization procedure by first applying SINDy for fixed delay and then subsequently optimizing the error of the reconstructed SINDy model over delay times. We test the SINDy-delay method on a noisy short data set from a toy delay differential equation and show excellent agreement. We then apply the method to experimental data of gene expressions in the bacterium…
Many dynamical systems of interest are nonlinear, with examples in turbulence, epidemiology, neuroscience, and finance, making them difficult to control using linear approaches. Model predictive control (MPC) is a powerful model-based optimization technique that enables the control of such nonlinear systems with constraints. However, modern systems often lack computationally tractable models, motivating the use of system identification techniques to learn accurate and efficient models for real-time control. In this tutorial article, we review emerging data-driven methods for model discovery an…
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）
PySINDy is a Python package for the discovery of governing dynamical systems models from data. In particular, PySINDy provides tools for applying the sparse identification of nonlinear dynamics (SINDy) (Brunton et al. 2016) approach to model discovery. In this work we provide a brief description of the mathematical underpinnings of SINDy, an overview and demonstration of the features implemented in PySINDy (with code examples), practical advice for users, and a list of potential extensions to PySINDy. Software is available at https://github.com/dynamicslab/pysindy.
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）
We propose a probabilistic model discovery method for identifying ordinary differential equations (ODEs) governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, in which target ODE models are expressed as a sparse linear combinations of pre-specified candidate functions. Promoting parsimony through sparsity in SINDy leads to interpretable models that generalize to unknown data. Instead of targeting point estimates of the SINDy (linear combination) coefficients, in this work we estimate these coefficients vi…
Sparse Identification of Nonlinear Dynamics (SINDy) has been shown to successfully recover governing equations from data; however, this approach assumes the initial condition to be exactly known in advance and is sensitive to noise. In this work we propose an integral SINDy (ISINDy) method to simultaneously identify model structure and parameters of nonlinear ordinary differential equations (ODEs) from noisy time-series observations. First, the states are estimated via penalized spline smoothing and then substituted into the integral-form numerical discretization solver, leading to a pseudo-li…
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…
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）
One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators
State Key Lab of Disaster Reduction in Civil Engineering（灾害防治ivil工程国家重点实验室）；Department of Civil and Environmental Engineering（土木与环境工程系）；Guangxi Laboratory of Whole Life Safety for Land-Sea Corridor Engineering（广西陆海通道工程全生命周期安全实验室）
We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stoc…
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）
Physiology-Informed Sparse Identification of Dynamic Crop Models
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Applied Statistics, Wageningen University and Research（瓦赫宁根大学与研究与应用统计系）
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）
Worcester Polytechnic Institute（伍斯特理工学院）；University of Bologna（博洛尼亚大学）；Smith College（史密斯学院）
We present a new algorithm for learning unknown governing equations from trajectory data, using and ensemble of neural networks. Given samples of solutions $x(t)$ to an unknown dynamical system $\dot{x}(t)=f(t,x(t))$, we approximate the function $f$ using an ensemble of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for $f$. This procedure yields M-1 time-independent networks, where M is the number of time steps at which $x(t)$ is observed. Finally…
Poincaré maps are an integral aspect to our understanding and analysis of nonlinear dynamical systems. Despite this fact, the construction of these maps remains elusive and is primarily left to simple motivating examples. In this manuscript we propose a method of data-driven discovery of Poincaré maps based upon sparse regression techniques, specifically the sparse identification of nonlinear dynamics (SINDy) algorithm. This work can be used to determine the dynamics on and near invariant manifolds of a given dynamical system, as well as provide long-time forecasting of the coarse-grained dyna…
