Sparse nonlinear models of chaotic electroconvection

Guan, Yifei; Brunton, Steven L.; Novosselov, Igor

doi:10.1098/rsos.202367

Cited by 29 publications

(12 citation statements)

References 112 publications

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“…The true system has ODEs: ẋ = −0.1x + 2y (9) ẏ = −2x − 0.1y (10) Three training trajectories had initial conditions [2, 0], [4,1], and [7,1]; and two holdout trajectories had initial conditions [3,2] and [6,3]. The initial conditions of the first training trajectory were from [1]; all others were selected at random.…”

Section: ) Linear Harmonic Oscillatormentioning

confidence: 99%

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Delahunt

Kutz

2022

IEEE Access

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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 ẋ = 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-complete library and yield a set of sparse equations that regress to the derivate ẋ. This toolkit includes: (regression step) weight timepoints based on estimated noise, use ensembles to estimate coefficients, and regress using FFTs; (culling step) leverage linear dependence of functionals, and restore and protect culled functionals based on Figures of Merit (FoMs). In a novel Assessment step, we define FoMs that compare model predictions to the original time-series (i.e., x(t) rather than ẋ(t)). These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g., 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1%−3% (for 50% noise), 6%−8% (for 100% noise); and 23%−25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on ẋ = f (x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.

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Section: ) Linear Harmonic Oscillatormentioning

confidence: 99%

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Delahunt

Kutz

2022

IEEE Access

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“…These models may be ordinary differential equations (ODEs) [16] or partial differential equations (PDEs) [17,29]. SINDy has been applied to a number of challenging model discovery problems, including for reduced-order models of fluid dynamics [30][31][32][33][34][35] and plasma dynamics [36][37][38], turbulence closures [39][40][41], mesoscale ocean closures [42], nonlinear optics [43], computational chemistry [44] and numerical integration schemes [45]. SINDy has been widely adopted, in part, because it is highly extensible.…”

Section: Introductionmentioning

confidence: 99%

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

et al. 2022

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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 the 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 forecasts. We apply this ensemble-SINDy (E-SINDy) algorithm to several synthetic and real-world datasets and demonstrate substantial improvements to the accuracy and robustness of model discovery from extremely noisy and limited data. For example, E-SINDy uncovers partial differential equations models from data with more than twice as much measurement noise as has been previously reported. Similarly, E-SINDy learns the Lotka Volterra dynamics from remarkably limited data of yearly lynx and hare pelts collected from 1900 to 1920. E-SINDy is computationally efficient, with similar scaling as standard SINDy. Finally, we show that ensemble statistics from E-SINDy can be exploited for active learning and improved model predictive control.

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“…We use the sparse identification of nonlinear dynamics (SINDy; Brunton et al 2016) method, which discovers equations that accurately reproduce the observed dynamics with as small a number of terms as possible. SINDy is now available as a Python module (de Silva et al 2020) and applied to numerous fields (e.g., Arzani & Dawson 2020;Guan et al 2021;Horrocks & Bauch 2020). Unlike three-dimensional hydrodynamical simulations, concise systems of governing equations are easily interpretable, similarly to those derived by a simple physical model.…”

Section: Introductionmentioning

confidence: 99%

Sparse Identification of Variable Star Dynamics

Pasquato

Abbas

Trani

et al. 2022

ApJ

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Variable stars play a crucial role as standard candles and provide valuable insights into stellar physics. They can be modeled either through fully fledged hydrodynamical simulations or analytically as systems of coupled differential equations describing the evolution of relevant physical quantities. Typically, such equations are arrived at by simplified physical assumptions concerning the conservation laws governing stellar interiors. Here we apply a data-driven technique—sparse identification of nonlinear dynamics (SINDy)—to automatically learn governing equations from observed light curves. We apply SINDy to 3100 light curves of three different variable types from the Catalina Sky Survey. The success rate depends systematically on variable type, with possible implications for variable star classification; however, it does not obviously depend on amplitude or period. Successful models can be reduced to the generalized Lienard equation x ̈ + ( a + bx + c x ̇ ) x ̇ + x = 0 . Members of the Lienard class of ordinary differential equations, such as the well-studied van der Pol oscillator, already saw some application to variable star modeling. For a, b = 0 the equation can be solved exactly, and it admits both periodic and nonperiodic solutions. We find a condition on the coefficients of the general equation for the presence of a limit cycle, which is also observed numerically in several instances.

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Sparse nonlinear models of chaotic electroconvection

Cited by 29 publications

References 112 publications

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control

Sparse Identification of Variable Star Dynamics

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