Although extensivework hasrecently been done inthisfield, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable.

The identification of ordinary differential equations (ODEs) and dynamical systems is a fundamental problem in control [32, 59, 60], data assimilation [42, 84], and more recently in scientific machine learning (ML) [11, 72, 74]. While algorithms such as Sparse Identification of Nonlinear Dynamics (SINDy) and its variants [46] are widely used by practitioners, they often fail in scenarios where observations of the state of the system are scarce, indirect, and noisy. In such scenarios modifications to SINDy-type methods are required to enforce additional constraints on the recovered equations to make them consistent with the observational data. Put simply, traditional SINDy-type methods work in two steps: (1) the data is used to filter the state of the system and estimate the derivatives, and (2) the filtered state is used to learn the underlying dynamics. In the regime of scarce, noisy and incomplete data, step 1 is inaccurate, which can propagate to poor results in the subsequent step 2. In this paper, we propose an all-at-once approach to filtering and equation learning based on collocation in a reproducing kernel Hilbert space (RKHS) which we term Joint SINDy (JSINDy), and shows that the issues above can be mitigated by performing both steps together. This joins a broader class of dynamics-informed methods that integrate the governing equations directly into the learning objective, either as hard constraints or as least-squares relaxations, which couples the problems of state estimation and model discovery. Representative examples include physics-informed and sparse-regression frameworks based on neural networks, splines, kernels, finite differences, and adjoint methods [21, 27, 39, 41, 72, 73, 88].

In this work we analyze the effectiveness of the Sparse Identification of Nonlinear Dynamics (SINDy) technique on three benchmark datasets for nonlinear identification, to provide a better understanding of its suitability when tackling real dynamical systems. While SINDy can be an appealing strategy for pursuing physics-based learning, our analysis highlights difficulties in dealing with unobserved states and non-smooth dynamics. Due to the ubiquity of these features in real systems in general, and control applications in particular, we complement our analysis with hands-on approaches to tackle these issues in order to exploit SINDy also in these challenging contexts.

Discovering the governing equations of evolving systems from available observations is essential and challenging. In this paper, we consider a new scenario: discovering governing equations from streaming data. Current methods struggle to discover governing differential equations with considering measurements as a whole, leading to failure to handle this task. We propose an online modeling method capable of handling samples one by one sequentially by modeling streaming data instead of processing the entire dataset. The proposed method performs well in discovering ordinary differential equations (ODEs) and partial differential equations (PDEs) from streaming data. Evolving systems are changing over time, which invariably changes with system status. Thus, finding the exact change points is critical. The measurement generated from a changed system is distributed dissimilarly to before; hence, the difference can be identified by the proposed method. Our proposal is competitive in identifying the change points and discovering governing differential equations in three hybrid systems and two switching linear systems.

Engineers have long relied on linearization to bridge the gap between simplified, linear descriptions where powerful analytical tools exist, and the intricate complexities of nonlinear dynamics where analytical solutions are elusive [5, 6]. Local linearization, implemented via first-order Taylor series approximation, has been widely used in system identification [5], optimization [6], and many other fields to make problems tractable. However, many real-world systems are fundamentally nonlinear and require solutions outside of the local neighborhood where linearization is valid. Rapid progress in machine learning and big data methods are driving advances in the data-driven modeling of such nonlinear systems in science and engineering [7] Koopman operator theory in particular has emerged as a principled approach to embed nonlinear dynamics in a linear framework that goes beyond simple linearization [4]. In the diverse landscape of data-driven modeling approaches, Koopman operator theory has received considerable attention in recent years [8-13]. These strategies encompass not only linear methodologies [5, 14] and dynamic mode decomposition (DMD) [1, 2, 15], but also more advanced techniques such as nonlinear autoregressive algorithms [16, 17], neural networks [18-27], Gaussian process regression [28], operator inference, and reduced-order modeling [29-31], among others [32-38]. The Koopman operator perspective is unique within data-driven modeling techniques due to its distinct aim of learning a coordinate system in which the nonlinear dynamics become linear. This methodology enables the application of closed-form, convergence-guaranteed methods from linear system theory to general nonlinear dynamics. To fully leverage the potential of data-driven Koopman theory across a diverse range of scientific and engineering disciplines, it is critical to have a central toolkit to automate state-of-the-art Koopman operator algorithms.

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-source tool can be used to quantitatively compare different methods of system identification. To illustrate how this benchmark can be utilized, we perform a large comparison of four algorithms for solving the sparse identification of nonlinear dynamics (SINDy) optimization problem, finding strong performance of the original algorithm and a recent mixed-integer discrete algorithm. In all cases, we used ensembling to improve the noise robustness of SINDy and provide statistical comparisons. In addition, we show very compelling evidence that the weak SINDy formulation provides significant improvements over the traditional method, even on clean data. Lastly, we investigate how Pareto-optimal models generated from SINDy algorithms depend on the properties of the equations, finding that the performance shows no significant dependence on a set of dynamical properties that quantify the amount of chaos, scale separation, degree of nonlinearity, and the syntactic complexity.