Big data have become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets, and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes.

Scientists have long quantified empirical observations by developing mathematical models that characterize the observations, have some measure of interpretability, and are capable of making predictions. Dynamical systems models in particular have been widely used to study, explain, and predict system behavior in a wide range of application areas, with examples ranging from Newton's laws of classical mechanics to the Michaelis-Menten kinetics for modeling enzyme kinetics. While governing laws and equations were traditionally derived by hand, the current growth of available measurement data and resulting emphasis on data-driven modeling motivates algorithmic approaches for model discovery. A number of such approaches have been developed in recent years and have generated widespread interest, including Eureqa (Schmidt & Lipson, 2009), sure independence screening and sparsifying operator (Ouyang, Curtarolo, Ahmetcik, Scheffler, & Ghiringhelli, 2018), and the sparse identification of nonlinear dynamics (SINDy) (Brunton, Proctor, & Kutz, 2016). Maximizing the impact of these model discovery methods requires tools to make them widely accessible to scientists across domains and at various levels of mathematical expertise.

We present new grids of Potsdam Wolf-Rayet (PoWR) model atmospheres for Wolf-Rayet stars of the nitrogen sequence (WN stars). The models have been calculated with the latest version of the PoWR stellar atmosphere code for spherical stellar winds. The WN model atmospheres include the non-LTE solutions of the statistical equations for complex model atoms, as well as the radiative transfer equation in the co-moving frame. Iron-line blanketing is treated with the help of the superlevel approach, while wind inhomogeneities are taken into account via optically thin clumps. Three of our model grids are appropriate for Galactic metallicity. The hydrogen mass fraction of these grids is 50%, 20%, and 0%, thus also covering the hydrogen-rich late-type WR stars that have been discovered in recent years. Three grids are adequate for LMC WN stars and have hydrogen fractions of 40%, 20%, and 0%. Recently, additional grids with SMC metallicity and with 60%, 40%, 20%, and 0% hydrogen have been added. We provide contour plots of the equivalent widths of spectral lines that are usually used for classification and diagnostics.

We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast.

Many inference problems relate to a dynamical system, as represented by dx/dt = f (x), where x ∈ ℝn is the state vector and f is the (in general nonlinear) system function or model. Since the time of Newton, researchers have pondered the problem of system identification: how should the user accurately and efficiently identify the model f – including its functional family or parameter values – from discrete time-series data? For linear models, many methods are available including linear regression, the Kalman filter and autoregressive moving averages. For nonlinear models, an assortment of machine learning tools have been developed in recent years, usually based on neural network methods, or various classification or order reduction schemes. The first group, while very useful, provide “black box" solutions which are not readily adaptable to new situations, while the second group necessarily involve the sacrificing of resolution to achieve order reduction. To address this problem, we propose the use of an inverse Bayesian method for system identification from time-series data. For a system represented by a set of basis functions, this is shown to be mathematically identical to Tikhonov regularization, albeit with a clear theoretical justification for the residual and regularization terms, respectively as the negative logarithms of the likelihood and prior functions. This insight justifies the choice of regularization method, and can also be extended to access the full apparatus of the Bayesian inverse solution. Two Bayesian methods, based on the joint maximum a posteriori (JMAP) and variational Bayesian approximation (VBA), are demonstrated for the Lorenz equation system with added Gaussian noise, in comparison to the regularization method of least squares regression with thresholding (the SINDy algorithm). The Bayesian methods are also used to estimate the variances of the inferred parameters, thereby giving the estimated model error, providing an important advantage of the Bayesian approach over traditional regularization methods.